set
stringclasses 1
value | id
stringlengths 5
9
| chunk_text
stringlengths 1
115k
| chunk_num_tokens
int64 1
106k
| document_num_tokens
int64 58
521k
| document_language
stringclasses 2
values |
|---|---|---|---|---|---|
train
|
0.35.10
|
\begin{subequations}\label{se:constructinvts2}
Here is a generalization. Suppose $G_1$ is a subgroup of $G$
normalized by $H$:
\begin{equation}
G_1 \subset G, \qquad \Ad(H)(G_1) \subset G_1.
\end{equation}
(The easiest way for this to happen is for $G_1$ to contain $H$.) Then
$H$ acts on ${\mathfrak Z}({\mathfrak g}_1)$, so we get
\begin{equation}\label{eq:Zg1i}
i_{G_1}\colon {\mathfrak Z}({\mathfrak g}_1)^H \rightarrow I^\tau(G/H),
\qquad z_1 \mapsto z_1\otimes I_{V_\tau}.
\end{equation}
These invariant differential operators are acting along the
submanifolds
\begin{equation}
xG_1/(G_1\cap H) \subset G/H \qquad (x\in G)
\end{equation}
of $G/H$. An example is the first coordinate $G_1=U(1)$ introduced in
\eqref{se:Ureps}, for $H=U(n-1)$. The operator $\Omega_{U(1)}$ on
$S^{2n-1}$ (acting along the fibers of the map $S^{2n-1} \rightarrow
{\mathbb C}{\mathbb P}^{n-1}$) is one of these new invariant
operators. A more interesting example is $G_1=Sp(1)\times Sp(1)$ studied
in \eqref{e:extraHreps}.
Here is how the spectral theory of these new operators is related to
representation theory. The map \eqref{eq:harmanalysis} is (by
Frobenius reciprocity) the same thing as an $H$-equivariant map
\begin{equation}
j_H\colon E_\pi \rightarrow V_\tau
\end{equation}
or equivalently
\begin{equation}
j_H^*\colon V_\tau^* \rightarrow E_\pi^*.
\end{equation}
It makes sense to define
\begin{equation}
(E_\pi^*)^{G_1,j_H} = \text{$G_1$ representation generated by
$j_H^*(V_\tau^*)$} \subset \pi^*.
\end{equation}
If the $G_1$ representation $(\pi^*)^{G_1,j_H}$ has infinitesimal character
$\chi_1^*$ (the contragredient of the infinitesimal character $\chi_1$),
then
\begin{equation} \label{eq:spectral2}
i_{G_1}(z_1) \text{\ acts on $j_G(E_\pi) \subset C^\infty({\mathcal V}_\tau)$ by
the scalar $\chi_1(z_1)$} \qquad (z_1 \in {\mathfrak Z}({\mathfrak
g}_1)^H).
\end{equation}
The homomorphisms $i_G$ of \eqref{eq:Zgi} and \eqref{eq:Zg1i} define an
algebra homomorphism from the abstract (commutative) tensor product
algebra
\begin{equation}\label{eq:Zgprodi}
i_G\otimes i_{G_1} \colon {\mathfrak Z}({\mathfrak g}) \otimes_{\mathbb
C}{\mathfrak Z}({\mathfrak g}_1) \rightarrow I^\tau(G/H).
\end{equation}
The reason for this is that ${\mathfrak Z}({\mathfrak g})$ commutes
with all of $U({\mathfrak g})$.
\end{subequations}
Now that we understand the relationship between representations in
$C^\infty({\mathcal V}_\tau)$ and the spectrum of invariant
differential operators, let us see what the results of Sections
\ref{sec:R}--\ref{sec:bigG2} can tell us: in particular, about the
kernel of the homomorphism $i_G\otimes i_{G_1}$ of \eqref{eq:Zgprodi}.
\begin{subequations}\label{se:Rdiff}
We begin with $G=O(n)$, $H=O(n-1)$ as in Section
\ref{sec:R}. Write $n=2m+\epsilon$, with $\epsilon=0$ or
$1$. A maximal torus in $G$ is
\begin{equation}
T=SO(2)^m,\qquad {\mathfrak t}_0 = {\mathbb
R}^m, \qquad {\mathfrak t} = {\mathbb C}^m.
\end{equation}
The Weyl group $W(O(n))$ acts by permutation and sign changes on
these $m$ coordinates. Harish-Chandra's theorem identifies
\begin{equation}\label{eq:HCO}
{\mathfrak Z}({\mathfrak g}) \simeq S({\mathfrak t})^{W(O(n))} =
{\mathbb C}[x_1,\cdots,x_m]^{W(O(n))}.
\end{equation}
Therefore
\begin{equation}
\hbox{(maximal ideals in ${\mathfrak Z}({\mathfrak g})$})
\leftrightarrow {\mathbb C}^m/W(O(n)).
\end{equation}
Suppose $z\in {\mathfrak Z}({\mathfrak g})$ corresponds to $p\in
{\mathbb C}[x_1,\cdots,x_m]^{W(O(n))}$ by \eqref{eq:HCO}. According to
\eqref{eq:spectral1} and \eqref{eq:Oinfchar}, the invariant
differential operator $i_G(z)$ will act on $\pi_a\subset
C^\infty(G/H)$ by the scalar
$$ p(a+(n-2)/2, (n-4)/2,\cdots,(n-2m)/2).$$
Recalling that $n-2m=\epsilon=0$ or $1$, we write this as
\begin{equation}\label{eq:Rscalar}
p(a+(n-2)/2, (n-4)/2,\cdots,\epsilon/2).
\end{equation}
\end{subequations}
Here is the consequence we want.
\begin{proposition}\label{prop:Rdiff} With notation as above,
the polynomial
$$p\in {\mathbb C}[x_1,\cdots,x_m]^{W(O(n))}$$
vanishes on the (affine) line
$$\{(\alpha,(n-4)/2,\cdots,\epsilon/2) \mid \alpha\in {\mathbb C}\}.$$
if and only if $i_G(z)\in I(G/H)$ is equal to zero.
\end{proposition}
\begin{proof} The statement ``if'' is a consequence of
\eqref{eq:Rscalar}: if the differential operator is zero, then $p$
must vanish at all the points $(a+(n-2)/2, (n-4)/2,\cdots)$ with $a$
a non-negative integer. These points are Zariski dense in the
line. For ``only if,'' the vanishing of the polynomial makes the
differential operator act by zero on all the subspaces $\pi_a\subset
C^\infty(G/H)$. The sum of these subspaces is dense (for example as
a consequence of \eqref{eq:Osphere}); so the differential operator
acts by zero. The faithfulness statement in Proposition
\ref{prop:invt} then implies that $i_G(z)=0$. \end{proof}
| 1,802 | 71,259 |
en
|
train
|
0.35.11
|
\begin{corollary}\label{cor:Rdiff}
\addtocounter{equation}{-1}
\begin{subequations}
The $O(n)$ infinitesimal characters
factoring to $i_G({\mathfrak Z}({\mathfrak g}))$ are indexed by
weights
\begin{equation}\label{eq:Rinfchar}
(\alpha,(n-4)/2,\cdots,\epsilon/2) \qquad (\alpha \in {\mathbb C}).
\end{equation}
Suppose $(\pi,E_\pi)$ is a representation of ${\mathfrak
o}(n,{\mathbb C})$ having an infinitesimal character, and that
$(E_\pi^*)^{{\mathfrak o}(n-1,{\mathbb C})} \ne 0$. Then $\pi$ has
infinitesimal character of the form \eqref{eq:Rinfchar}.
\end{subequations}
\end{corollary}
Exactly the same arguments apply to the other examples treated in
Sections \ref{sec:R}--\ref{sec:bigG2}. We will just state the
conclusions.
\begin{subequations}\label{se:Cdiff}
Suppose $G=U(n)$, $H=U(n-1)$ as in Section
\ref{sec:C}. A maximal torus in $G$ is
\begin{equation}
T=U(1)^n,\qquad {\mathfrak t}_0 = {\mathbb
R}^n, \qquad {\mathfrak t} = {\mathbb C}^n.
\end{equation}
The Weyl group $W(U(n))$ acts by permutation on
these $n$ coordinates. Harish-Chandra's theorem identifies
\begin{equation}\label{eq:HCU}
{\mathfrak Z}({\mathfrak g}) \simeq S({\mathfrak t})^{W(U(n))} =
{\mathbb C}[x_1,\cdots,x_n]^{W(U(n))}.
\end{equation}
Therefore
\begin{equation}
\hbox{(maximal ideals in ${\mathfrak Z}({\mathfrak g})$})
\leftrightarrow {\mathbb C}^n/W(U(n)).
\end{equation}
Suppose $z\in {\mathfrak Z}({\mathfrak g})$ corresponds to $p\in
{\mathbb C}[x_1,\cdots,x_n]^{W(U(n))}$ by \eqref{eq:HCU}. According to
\eqref{eq:spectral1} and \eqref{eq:Uinflchar}, the invariant
differential operator $i_G(z)$ will act on $\pi_{b,c}\subset
C^\infty(G/H)$ by the scalar
\begin{equation}\label{eq:Cscalar}
p((b+(n-1))/2, (n-3)/2,\cdots,-(n-3)/2,-(c+(n-1))/2).
\end{equation}
\end{subequations}
\begin{proposition}\label{prop:Cdiff} With notation as above,
the polynomial
$$p\in {\mathbb C}[x_1,\cdots,x_n]^{W(U(n))}$$
vanishes on the (affine) plane
$$\{(\xi,(n-3)/2,\cdots,-(n-3)/2,-\tau) \mid (\xi,\tau)\in {\mathbb C}^2\}.$$
if and only if $i_G(z)\in I(G/H)$ is equal to zero.
\end{proposition}
\begin{corollary}\label{cor:Cdiff}
\addtocounter{equation}{-1}
\begin{subequations}
The $U(n)$ infinitesimal characters
factoring to $i_G({\mathfrak Z}({\mathfrak g}))$ are indexed by
weights
\begin{equation}\label{eq:Cinfchar}
(\xi,(n-3)/2,\cdots,-(n-3)/2,-\tau) \qquad ((\xi,\tau) \in
{\mathbb C}^2).
\end{equation}
Suppose $(\gamma,F_\gamma)$ is a representation of ${\mathfrak
u}(n,{\mathbb C})$ having an infinitesimal character, and that
$(F_\gamma^*)^{{\mathfrak u}(n-1,{\mathbb C})} \ne 0$. Then
$F_\gamma$ has infinitesimal character of the form \eqref{eq:Cinfchar}. The
parameters $\xi$ and $\tau$ may be determined as follows. The
central character of $\gamma$ (scalars by which the one-dimensional center of
the Lie algebra acts) is given by $\xi-\tau$. If in addition
$F_\gamma \subset E_\pi$ for
some representation $(\pi,E_\pi)$ of ${\mathfrak o}(2n,{\mathbb C})$
as in Corollary \ref{cor:Rdiff}, then we may take $\xi+\tau =
\alpha$. (Replacing $\alpha$ by the equivalent infinitesimal
character parameter $-\alpha$
has the effect of interchanging $\xi$ and $-\tau$, which defines an
equivalent infinitesimal character parameter.)
\end{subequations}
\end{corollary}
\begin{subequations}\label{se:Hdiff}
Suppose next that $G=Sp(n)\times Sp(1)$, $H=Sp(n-1)\times
Sp(1)_\Delta$ as in Section \ref{sec:H}. A maximal torus in $G$ is
\begin{equation}
T=U(1)^n\times U(1),\qquad {\mathfrak t}_0 = {\mathbb
R}^n\times {\mathbb R}, \qquad {\mathfrak t} = {\mathbb C}^n
\times {\mathbb C}.
\end{equation}
The Weyl group $W(Sp(n)\times Sp(1))$ acts by sign changes on all
$n+1$ coordinates, and permutation of the first $n$
coordinates. Harish-Chandra's theorem identifies
\begin{equation}\label{eq:HCSp}
{\mathfrak Z}({\mathfrak g}) \simeq S({\mathfrak t})^{W(Sp(n)\times Sp(1))} =
{\mathbb C}[x_1,\cdots,x_n,y]^{W(Sp(n)\times Sp(1))}.
\end{equation}
Therefore
\begin{equation}
\hbox{(maximal ideals in ${\mathfrak Z}({\mathfrak g})$})
\leftrightarrow {\mathbb C}^{n+1}/W(Sp(n)\times Sp(1)).
\end{equation}
Suppose $z\in {\mathfrak Z}({\mathfrak g})$ corresponds to $p\in
{\mathbb C}[x_1,\cdots,x_n,y]^{W(Sp(n)\times Sp(1))}$ by
\eqref{eq:HCSp}. According to \eqref{eq:spectral1} and
\eqref{eq:Hinflchar}, the invariant differential operator $i_G(z)$
will act on $\pi_{d,e}\subset C^\infty(G/H)$ by the scalar
\begin{equation}\label{eq:Hscalar}
p((d+n, e+(n-1),n-2,\cdots,1),(d-e+1)).
\end{equation}
\end{subequations}
\begin{proposition}\label{prop:Hdiff} With notation as above,
the polynomial
$$p\in {\mathbb C}[x_1,\cdots,x_n,y]^{W(Sp(n)\times Sp(1))}$$
vanishes on the (affine) plane
$$\{(\xi,\tau,n-2,\cdots,1)(\xi-\tau) \mid (\xi,\tau)\in {\mathbb C}^2\}.$$
if and only if $i_G(z)\in I(G/H)$ is equal to zero.
\end{proposition}
\begin{corollary}\label{cor:Hdiff}
\addtocounter{equation}{-1}
\begin{subequations}
The infinitesimal characters for
$Sp(n)\times Sp(1)$ which factor to $i_G({\mathfrak Z}({\mathfrak
g}))$ are indexed by weights
\begin{equation}\label{eq:Hinfchar}
(\xi,\tau,n-2,\cdots,1)(\xi-\tau) \qquad ((\xi,\tau) \in {\mathbb
C}^2).
\end{equation}
Suppose $(\gamma,F_\gamma)$ is a representation of ${\mathfrak
s}{\mathfrak p}(n,{\mathbb C}) \times {\mathfrak
s}{\mathfrak p}(1,{\mathbb C})$ having an infinitesimal
character, and that $(F_\gamma^*)^{{\mathfrak s}{\mathfrak
p}(n-1,{\mathbb C}) \times {\mathfrak
s}{\mathfrak p}(1,{\mathbb C})_\Delta} \ne 0$. Then $F_\gamma$
has infinitesimal character of the form \eqref{eq:Hinfchar};
$\xi-\tau$ is the infinitesimal character of the ${\mathfrak
s}{\mathfrak p}(1,{\mathbb C})$ factor. If in
addition $F_\gamma \subset
E_\pi$ for some representation $(\pi,E_\pi)$ of ${\mathfrak
o}(4n,{\mathbb C})$ as in Corollary \ref{cor:Rdiff}, then we may take
$\xi+\tau = \alpha$.
\end{subequations}
\end{corollary}
This is a good setting in which to consider the more general invariant
differential operators from \eqref{se:constructinvts2}.
\begin{subequations}\label{se:HG1diff}
Suppose in that general setting that $G_1$ is reductive, and choose a Cartan
subalgebra ${\mathfrak t}_1\subset {\mathfrak g}_1$, with (finite) Weyl group
\begin{equation}
W(G_1) =_{\text{def}} N_{G_1({\mathbb C})}({\mathfrak
t}_1)/Z_{G_1({\mathbb C})}({\mathfrak t_1}) \subset
\Aut({\mathfrak t}_1), \qquad {\mathfrak Z}({\mathfrak g}_1)
\simeq S({\mathfrak t})^{W_1}.
\end{equation}
The adjoint action of $H$ on $G_1$ defines another Weyl group,
which normalizes $W(G_1)$:
\begin{equation}
W(G_1) \triangleleft W_H(G_1) =_{\text{def}} N_{H({\mathbb C})}({\mathfrak
t}_1)/Z_{H({\mathbb C})}({\mathfrak t_1}) \subset \Aut({\mathfrak
t}_1), \qquad {\mathfrak Z}({\mathfrak g}_1)^H
\simeq S({\mathfrak t}_1)^{W_H(G_1)}.
\end{equation}
Under mild hypotheses (for example $G_1$ is reductive algebraic and the
adjoint action of $H$ is algebraic) then $W_H(G_1)$ is finite, so the
algebra ${\mathfrak Z}({\mathfrak g}_1)$ is finite over ${\mathfrak
Z}({\mathfrak g}_1)^H$, and the maximal ideals in this smaller algebra
are given by evaluation at
\begin{equation}
\mu\in {\mathfrak t}_1^*/W_H(G_1).
\end{equation}
In the case $G_1=Sp(1)\times Sp(1)$, the adjoint action of $H$ on $G_1$ is
contained in that of $G_1$, so $W(G_1)=W_H(G_1)$, and ${\mathfrak Z}({\mathfrak
g}_1)^H = {\mathfrak Z}({\mathfrak g}_1)$. We have
\begin{equation}
T_1=U(1)^2, \qquad {\mathfrak t}_{1,0} = {\mathbb R}^2, \qquad {\mathfrak
t}_1 = {\mathbb C}^2.
\end{equation}
The Weyl group $W(G_1)=W_H(G_1)$ acts by sign changes on each
coordinate, so the Harish-Chandra isomorphism is
\begin{equation}\label{eq:HCH}
{\mathfrak Z}({\mathfrak g}_1)^H = {\mathfrak Z}({\mathfrak g}_1) \simeq
S({\mathfrak t}_1)^{W(G_1)} = {\mathbb C}[u_1,u_2]^{W(G_1)}
\end{equation}
Suppose therefore that $z_1\in {\mathfrak Z}({\mathfrak g}_1)$
corresponds to $p_1\in {\mathbb C}[x_1,x_2]^{W(G_1)}$. According to
\eqref{eq:spectral2} and
\eqref{eq:Hsubinflchar}, the invariant differential operator $i_{G_1}(z_1)$
acts on $\pi^{Sp(n)\times Sp(1)}_{d,e} \subset C^\infty(G/H)$ by the
scalar
\begin{equation}\label{eq:HG1scalar}
p_1(d-e+1,d-e+1).
\end{equation}
\end{subequations}
\begin{proposition}\label{prop:HG1diff} With notation as above,
suppose that
$$P\in {\mathbb C}[x_1,\cdots,x_n,y,u_1,u_2]^{W(G)\times W(G_1)},$$
and write $Z\in {\mathfrak Z}({\mathfrak g})\otimes {\mathfrak
Z}({\mathfrak g}_1)^H$ for the corresponding central element. Then $P$
vanishes on the affine plane
$$\{(\xi,\tau,n-2,\cdots,1)(\xi-\tau)(\xi-\tau,\xi-\tau)\mid
(\xi,\tau)\in {\mathbb C}^2\}.$$
if and only if $(i_G\otimes i_{G_1})(Z)\in I(G/H)$ is equal to zero.
\end{proposition}
\begin{corollary}\label{cor:HG1diff}
\addtocounter{equation}{-1}
\begin{subequations}
In the setting $G/H =
(Sp(n)\times Sp(1))/(Sp(n-1)\times Sp(1)_\Delta)$, $G_1=Sp(1)\times Sp(1)$,
the characters of the tensor product algebra \eqref{eq:Zgprodi}
which factor to the image in $I(G/H)$ are indexed by weights
\begin{equation}\label{eq:HG1infchar}
(\xi,\tau,n-2,\cdots,1)(\xi-\tau)(\xi-\tau,\xi-\tau).
\end{equation}
Here the first $n$ coordinates are giving the infinitesimal
character for $Sp(n)$; the next is the infinitesimal character for
the $Sp(1)$ factor of $G$; and the last two are the infinitesimal
character for $G_1$.
Suppose $(\gamma,F_\gamma)$ is an ${\mathfrak
s}{\mathfrak p}(n,{\mathbb C})$ representation as in Corollary
\ref{cor:Hdiff}. Then the ${\mathfrak g}_1$ representation
generated by $(F_\gamma^*)^{{\mathfrak s}{\mathfrak p}(n-1,{\mathbb
C})}$ has infinitesimal character $(\xi-\tau,\xi-\tau)$.
\end{subequations}
\end{corollary}
| 3,718 | 71,259 |
en
|
train
|
0.35.12
|
This is a good setting in which to consider the more general invariant
differential operators from \eqref{se:constructinvts2}.
\begin{subequations}\label{se:HG1diff}
Suppose in that general setting that $G_1$ is reductive, and choose a Cartan
subalgebra ${\mathfrak t}_1\subset {\mathfrak g}_1$, with (finite) Weyl group
\begin{equation}
W(G_1) =_{\text{def}} N_{G_1({\mathbb C})}({\mathfrak
t}_1)/Z_{G_1({\mathbb C})}({\mathfrak t_1}) \subset
\Aut({\mathfrak t}_1), \qquad {\mathfrak Z}({\mathfrak g}_1)
\simeq S({\mathfrak t})^{W_1}.
\end{equation}
The adjoint action of $H$ on $G_1$ defines another Weyl group,
which normalizes $W(G_1)$:
\begin{equation}
W(G_1) \triangleleft W_H(G_1) =_{\text{def}} N_{H({\mathbb C})}({\mathfrak
t}_1)/Z_{H({\mathbb C})}({\mathfrak t_1}) \subset \Aut({\mathfrak
t}_1), \qquad {\mathfrak Z}({\mathfrak g}_1)^H
\simeq S({\mathfrak t}_1)^{W_H(G_1)}.
\end{equation}
Under mild hypotheses (for example $G_1$ is reductive algebraic and the
adjoint action of $H$ is algebraic) then $W_H(G_1)$ is finite, so the
algebra ${\mathfrak Z}({\mathfrak g}_1)$ is finite over ${\mathfrak
Z}({\mathfrak g}_1)^H$, and the maximal ideals in this smaller algebra
are given by evaluation at
\begin{equation}
\mu\in {\mathfrak t}_1^*/W_H(G_1).
\end{equation}
In the case $G_1=Sp(1)\times Sp(1)$, the adjoint action of $H$ on $G_1$ is
contained in that of $G_1$, so $W(G_1)=W_H(G_1)$, and ${\mathfrak Z}({\mathfrak
g}_1)^H = {\mathfrak Z}({\mathfrak g}_1)$. We have
\begin{equation}
T_1=U(1)^2, \qquad {\mathfrak t}_{1,0} = {\mathbb R}^2, \qquad {\mathfrak
t}_1 = {\mathbb C}^2.
\end{equation}
The Weyl group $W(G_1)=W_H(G_1)$ acts by sign changes on each
coordinate, so the Harish-Chandra isomorphism is
\begin{equation}\label{eq:HCH}
{\mathfrak Z}({\mathfrak g}_1)^H = {\mathfrak Z}({\mathfrak g}_1) \simeq
S({\mathfrak t}_1)^{W(G_1)} = {\mathbb C}[u_1,u_2]^{W(G_1)}
\end{equation}
Suppose therefore that $z_1\in {\mathfrak Z}({\mathfrak g}_1)$
corresponds to $p_1\in {\mathbb C}[x_1,x_2]^{W(G_1)}$. According to
\eqref{eq:spectral2} and
\eqref{eq:Hsubinflchar}, the invariant differential operator $i_{G_1}(z_1)$
acts on $\pi^{Sp(n)\times Sp(1)}_{d,e} \subset C^\infty(G/H)$ by the
scalar
\begin{equation}\label{eq:HG1scalar}
p_1(d-e+1,d-e+1).
\end{equation}
\end{subequations}
\begin{proposition}\label{prop:HG1diff} With notation as above,
suppose that
$$P\in {\mathbb C}[x_1,\cdots,x_n,y,u_1,u_2]^{W(G)\times W(G_1)},$$
and write $Z\in {\mathfrak Z}({\mathfrak g})\otimes {\mathfrak
Z}({\mathfrak g}_1)^H$ for the corresponding central element. Then $P$
vanishes on the affine plane
$$\{(\xi,\tau,n-2,\cdots,1)(\xi-\tau)(\xi-\tau,\xi-\tau)\mid
(\xi,\tau)\in {\mathbb C}^2\}.$$
if and only if $(i_G\otimes i_{G_1})(Z)\in I(G/H)$ is equal to zero.
\end{proposition}
\begin{corollary}\label{cor:HG1diff}
\addtocounter{equation}{-1}
\begin{subequations}
In the setting $G/H =
(Sp(n)\times Sp(1))/(Sp(n-1)\times Sp(1)_\Delta)$, $G_1=Sp(1)\times Sp(1)$,
the characters of the tensor product algebra \eqref{eq:Zgprodi}
which factor to the image in $I(G/H)$ are indexed by weights
\begin{equation}\label{eq:HG1infchar}
(\xi,\tau,n-2,\cdots,1)(\xi-\tau)(\xi-\tau,\xi-\tau).
\end{equation}
Here the first $n$ coordinates are giving the infinitesimal
character for $Sp(n)$; the next is the infinitesimal character for
the $Sp(1)$ factor of $G$; and the last two are the infinitesimal
character for $G_1$.
Suppose $(\gamma,F_\gamma)$ is an ${\mathfrak
s}{\mathfrak p}(n,{\mathbb C})$ representation as in Corollary
\ref{cor:Hdiff}. Then the ${\mathfrak g}_1$ representation
generated by $(F_\gamma^*)^{{\mathfrak s}{\mathfrak p}(n-1,{\mathbb
C})}$ has infinitesimal character $(\xi-\tau,\xi-\tau)$.
\end{subequations}
\end{corollary}
\begin{subequations}\label{se:octdiff}
Suppose next that $G=\Spin(9)$, $H=\Spin(7)'$ as in Section
\ref{sec:O}. A maximal torus in $G$ is
\begin{equation}
T= \hbox{double cover of\ }SO(2)^4,\qquad {\mathfrak t}_0 = {\mathbb
R}^4, \qquad {\mathfrak t} = {\mathbb C}^4.
\end{equation}
The Weyl group $W(\Spin(9))$ acts by permutation and sign changes on these
four coordinates. Harish-Chandra's theorem identifies
\begin{equation}\label{eq:HCoct}
{\mathfrak Z}({\mathfrak g}) \simeq S({\mathfrak t})^{W(\Spin(9))} =
{\mathbb C}[x_1,\cdots,x_4]^{W(\Spin(9))}.
\end{equation}
Therefore
\begin{equation}
\hbox{(maximal ideals in ${\mathfrak Z}({\mathfrak g})$})
\longleftrightarrow {\mathbb C}^{4}/W(\Spin(9)).
\end{equation}
Suppose $z\in {\mathfrak Z}({\mathfrak g})$ corresponds to $p\in
{\mathbb C}[x_1,\cdots,x_4]^{W(\Spin(9))}$ by
\eqref{eq:HCoct}. According to \eqref{eq:spectral1} and
\eqref{eq:octinflchar}, the invariant differential operator $i_G(z)$
will act on $\pi^{\Spin(9)}_{x,y}\subset C^\infty(G/H)$ by the scalar
\begin{equation}\label{eq:octscalar}
p((2x+y+7)/2,(y+5)/2,(y+3)/2,(y+1)/2).
\end{equation}
\end{subequations}
\begin{proposition}\label{prop:octdiff} With notation as above,
the polynomial
$$p\in {\mathbb C}[x_1,\cdots,x_4]^{W(\Spin(9))}$$
vanishes on the (affine) plane
$$\{(\xi,\tau+5/2,\tau+3/2,\tau+1/2) \mid (\xi,\tau)\in
{\mathbb C}^2\}.$$
if and only if $i_G(z)\in I(G/H)$ is equal to zero.
\end{proposition}
\begin{corollary}\label{cor:octdiff}
\addtocounter{equation}{-1}
\begin{subequations}
The infinitesimal characters for
$\Spin(9)$ factoring to $i_G({\mathfrak Z}({\mathfrak
g}))$ are indexed by weights
\begin{equation}\label{eq:octinfchar}
(\xi,\tau+5/2,\tau+3/2,\tau+1/2) \qquad ((\xi,\tau) \in {\mathbb C}^2).
\end{equation}
Suppose $(\gamma,F_\gamma)$ is a representation of ${\mathfrak
s}{\mathfrak p}{\mathfrak i}{\mathfrak n}(9,{\mathbb C})$ having
an infinitesimal character, and that
$(F_\gamma^*)^{{\mathfrak h}({\mathbb C})} \ne 0$. Then $F_\gamma$ has
infinitesimal character of the form \eqref{eq:Hinfchar}. If the
${\mathfrak s} {\mathfrak p} {\mathfrak i} {\mathfrak n} (8,{\mathbb
C})$-module generated by $(F_\gamma^*)^{{\mathfrak h}({\mathbb
C})}$ has a submodule with an infinitesimal character, then we
may choose $\tau$ so that this infinitesimal character is
\begin{equation}
(\tau+3,\tau+2,\tau+1,\tau).
\end{equation}
If in addition $F_\gamma \subset E_\pi$ for
some representation $(\pi,E_\pi)$ of ${\mathfrak o}(16,{\mathbb C})$
as in Corollary \ref{cor:Rdiff} (with infinitesimal character
parameter $\alpha$) then we may choose $\xi=\alpha/2$.
\end{subequations}
\end{corollary}
For the last two cases we write even less.
\begin{corollary}\label{cor:G2diff}
\addtocounter{equation}{-1}
\begin{subequations}
When $G/H = G_{2,c}/SU(3)$, the
infinitesimal characters for
$G_{2,c}$ which factor to $i_G({\mathfrak Z}({\mathfrak
g}))$ are indexed by weights
\begin{equation}\label{eq:G2infchar}
(2\xi,(1/2)-\xi,-(1/2)-\xi) \qquad (\xi \in {\mathbb C}).
\end{equation}
Suppose $(\gamma,F_\gamma)$ is a representation of ${\mathfrak
g}_2({\mathbb C})$ having an infinitesimal character, and that
$(F_\gamma^*)^{{\mathfrak u}(3,{\mathbb C})} \ne 0$. Then the
infinitesimal character of $F_\gamma$ is of the form in
\eqref{eq:G2infchar}. If in addition $F_\gamma \subset E_\pi$ for
some representation $(\pi,E_\pi)$ of ${\mathfrak o}(7,{\mathbb C})$
as in Corollary \ref{cor:Rdiff}, then we may take $\xi = \alpha/3$.
\end{subequations}
\end{corollary}
\begin{corollary}\label{cor:bigG2diff}
\addtocounter{equation}{-1}
\begin{subequations}
When $G/H = \Spin(7)'/G_{2,c}$, the
infinitesimal characters for
$\Spin(7)'$ which factor to $i_G({\mathfrak Z}({\mathfrak
g}))$ are indexed by weights
\begin{equation}\label{eq:bigG2infchar}
(\xi+1,\xi,\xi-1) \qquad (\xi \in {\mathbb C}).
\end{equation}
Suppose $(\gamma,F_\gamma)$ is a representation of ${\mathfrak
s}{\mathfrak p}{\mathfrak i}{\mathfrak n}(7,{\mathbb C})'$ having
an infinitesimal character, and that
$(F_\gamma^*)^{{\mathfrak g}_2({\mathbb C})} \ne 0$. Then the
infinitesimal character of $F_\gamma$ is of the form in
\eqref{eq:bigG2infchar}. If in addition $F_\gamma \subset E_\pi$ for
some representation $(\pi,E_\pi)$ of ${\mathfrak o}(8,{\mathbb C})$
as in Corollary \ref{cor:Rdiff}, then we may take $\xi= \alpha/2$.
\end{subequations}
\end{corollary}
| 3,118 | 71,259 |
en
|
train
|
0.35.13
|
\section{Changing real forms}
\label{sec:changereal}
\setcounter{equation}{0}
Results like \eqref{eq:spectral1} and its generalization
\eqref{eq:spectral2} explain why it is interesting to study the
representations of $G$ appearing in $C^\infty({\mathcal V}_\tau)$ and
the invariant differential operators on this space. In this section we
state our first method for doing that.
\begin{definition}\label{def:realform}
\addtocounter{equation}{-1}
\begin{subequations}\label{se:realform}
Suppose $G_1$ and $G_2$ are Lie
groups with closed subgroups $H_1$ and $H_2$. Assume that there is
an isomorphism of complexified Lie algebras
\begin{equation}i\colon {\mathfrak g}_1 \buildrel{\sim}\over{\longrightarrow}
{\mathfrak g}_2, \qquad i({\mathfrak h}_1) = {\mathfrak h}_2.
\end{equation}
Finally, assume that $i$ identifies the Zariski closure of $\Ad(H_1)$
in $\Aut({\mathfrak g}_1)$ with the Zariski closure of $\Ad(H_2)$ in
$\Aut({\mathfrak g}_2)$. (This is automatic if $H_1$ and $H_2$ are
connected.) Then we say that the homogeneous space $G_2/H_2$ is {\em
another real form} of the homogeneous space $G_1/H_1$.
Given representations $(\tau_i,V_{\tau_i})$ of $H_i$, we say that
${\mathcal V}_{\tau_2}$ is {\em another real form} of ${\mathcal
V}_{\tau_1}$ if there is an isomorphism
\begin{equation}i\colon V_{\tau_1} \buildrel{\sim}\over{\longrightarrow}
V_{\tau_2}\end{equation}
respecting the actions of ${\mathfrak h}$, and identifying the
Zariski closure of $\Ad(H_1)$ in $\End(V_{\tau_1})$ with the Zariski
closure of $\Ad(H_2)$ in $\End(V_{\tau_2})$.
Whenever ${\mathcal V}_{\tau_2}$ is another real form of ${\mathcal
V}_{\tau_1}$, we get an algebra isomorphism
\begin{equation}\label{eq:realformisom}
i\colon {\mathbb D}^{\tau_1}(G_1/H_1)
\buildrel{\sim}\over{\longrightarrow} {\mathbb D}^{\tau_2}(G_2/H_2).
\end{equation}
\end{subequations}
\end{definition}
We will use these isomorphisms together with results like Corollaries
\ref{cor:Cdiff}--\ref{cor:bigG2diff} (proven using compact homogeneous
spaces $G_1/H_1$) to control the possible representations appearing in
some noncompact homogeneous spaces $G_2/H_2$.
\section{Changing the size of the group}
\label{sec:size}
\setcounter{equation}{0}
Our second way to study representations and invariant differential
operators is this.
\begin{subequations}\label{se:size}
In the setting \eqref{se:invt}, suppose that $S\subset G$ is a closed
subgroup, and that
\begin{equation}\label{eq:bigsub1}
\dim G/H = \dim S/(S\cap H).
\end{equation}
Equivalent requirements are
\begin{equation}\label{eq:bigsub2}
{\mathfrak s}/({\mathfrak s}\cap {\mathfrak h}) = {\mathfrak
g}/{\mathfrak h}
\end{equation}
or
\begin{equation}\label{eq:bigsub3}
{\mathfrak s} + {\mathfrak h} = {\mathfrak g}
\end{equation}
or
\begin{equation}\label{eq:bigsub4}
\hbox{$S/(S\cap H)$ is open in $G/H$}.
\end{equation}
Because of this open embedding, differential operators on $S/(S\cap
H)$ are more or less the same thing as differential operators on
$G/H$. The condition of $S$-invariance is weaker than the condition of
$G$-invariance, so we get natural inclusions
\begin{equation}\label{eq:opincl}
{\mathbb D}(G/H) \hookrightarrow {\mathbb D}(S/(S\cap H)), \qquad {\mathbb
D}^\tau(G/H) \hookrightarrow {\mathbb D}^\tau(S/(S\cap H)).
\end{equation}
(notation as in \eqref{se:invt}). In terms of the
algebraic description of these operators given in Proposition
\ref{prop:invt}, notice first that the condition in \eqref{eq:bigsub2}
shows that the inclusion ${\mathfrak s} \hookrightarrow {\mathfrak g}$
defines an isomorphism
\begin{equation}
U({\mathfrak s})\otimes_{{\mathfrak s}\cap{\mathfrak h}}\End(V_\tau)
\simeq U({\mathfrak g})\otimes_{{\mathfrak h}}\End(V_\tau)
\end{equation}
Therefore
\begin{equation}\begin{aligned}
\ [U({\mathfrak g})\otimes_{{\mathfrak
h}}\End(V_\tau)]^{(\Ad\otimes \Ad)(H)} &\hookrightarrow
[U({\mathfrak g})\otimes_{{\mathfrak
h}}\End(V_\tau)]^{(\Ad\otimes\Ad)(S\cap H)} \\
&\simeq [U({\mathfrak s})\otimes_{{\mathfrak s}\cap{\mathfrak
h}}\End(V_\tau)]^{(\Ad\otimes\Ad)(S\cap H)}.
\end{aligned} \end{equation}
That is,
\begin{equation}\label{eq:algincl}
I^\tau(G/H) \hookrightarrow I^\tau(S/(S\cap H)).
\end{equation}
This algebra inclusion corresponds to the differential operator
inclusion \eqref{eq:opincl} under the identification of Proposition
\ref{prop:invt}.
\end{subequations}
Here is a useful fact.
\begin{proposition} Let $G$ be a connected reductive Lie group,
and let $H$ and $S$ be closed connected reductive subgroups.
Assume the equivalent conditions (\ref{eq:bigsub1})-(\ref{eq:bigsub4}). Then
\begin{enumerate}
\item $G=SH$, and
\item there is a Cartan involution for $G$ preserving both $S$ and
$H$.
\end{enumerate}
\end{proposition}
\begin{proof} Part (1) is due to Onishchik \cite{Oni69}*{Theorem 3.1}.
For (2), since $H$ is reductive in $G$, there is a Cartan
involution $\theta_H$ for $G$ preserving $H$, and likewise there is
one $\theta_S$ preserving $S$. By the uniqueness of Cartan involutions
for $G$, $\theta_S$ is the conjugate of $\theta_H$ by some element $g\in G$,
which by (1) can be decomposed as $g=sh$. The $h$-conjugate of $\theta_H$,
which is also the $s^{-1}$-conjugate of $\theta_S$, has the required property.
\end{proof}
It follows from (1) that if $(G_c,S_c,H_c)$ is a triple of a compact Lie group
and two closed subgroups such that $G_c=S_cH_c$, and if $(G,S,H)$ is a triple
of real forms (that is, $G/S$ is a real form of $G_c/S_c$ and
$G/H$ a real form of $G_c/H_c$), then $S$ acts transitively on $G/H$.
Conversely, by (2) every transitive action on a reductive homogeneous space
$G/H$ by a reductive subgroup $S\subset G$ is obtained in this way.
In the following sections we shall apply this principle to the real hyperboloid
(\ref{eq:Hpq}), which is a real form of $S^{p+q-1}=O(p+q)/O(p+q-1)$.
The hypothesis that both $S$ and $H$ be reductive is certainly
necessary. Suppose for example that $S$ is a noncompact
real form of the complex reductive group $G$, and that
$H$ is a parabolic subgroup of $G$ (so that $S$ and $G$ are reductive,
but $H$ is not). Then $S$ has finitely many orbits on $G/H$
(\cite{Wolfflag}), and in particular has open orbits (so that the
conditions \eqref{eq:bigsub1}--\eqref{eq:bigsub4} are satisfied); but
the number of orbits is almost always greater than one (so $G \ne SH$).
| 2,191 | 71,259 |
en
|
train
|
0.35.14
|
\section{Classical hyperboloids}
\label{sec:Opq}
\setcounter{equation}{0}
\begin{subequations}\label{se:Opq}
In this section we recall the classical representation-theoretic
decomposition of functions on real hyperboloids: that is, on other
real forms of spheres. The spaces are
\begin{equation}\label{eq:Hpq}
H_{p,q} = \{v\in {\mathbb R}^{p,q} \mid \langle
v,v\rangle_{p,q} = 1\} = O(p,q)/O(p-1,q).
\end{equation}
Here $\langle,\rangle_{p,q}$ is the standard quadratic form of
signature $(p,q)$ on ${\mathbb R}^{p+q}$. The inclusion of the right
side of the equality in the middle is just given by the action of the
orthogonal group on the basis vector $e_1$; surjectivity is Witt's
theorem. This realization of the hyperboloid is a symmetric space, so
the Plancherel decomposition is completely known. In particular, the
discrete series may be described as follows. To avoid degenerate
cases, we assume that
\begin{equation}\label{eq:Obigp}
p \ge 2.
\end{equation}
There is a ``compact Cartan subspace'' with Lie algebra
\begin{equation}\label{eq:cptCartanOpq}
{\mathfrak a}_c = \langle e_{12} - e_{21} \rangle.
\end{equation}
The first requirement is that
\begin{equation}
{\mathfrak a}_c \subset {\mathfrak k} = {\mathfrak o}(p) \times
{\mathfrak o}(q).
\end{equation}
That this is satisfied is a consequence of \eqref{eq:Obigp}. The
second requirement is that ${\mathfrak a}_c$ belongs to the $-1$
eigenspace of the involutive automorphism
\begin{equation}
\sigma = \Ad\left(\diag(-1,1,1,\dots,1) \right)
\end{equation}
with fixed points the isotropy subgroup $O(p-1,q)$. (More precisely,
the group of fixed points of $\sigma$ is $O(1) \times O(p-1,q)$; so
our hyperboloid is a $2$-to-$1$ cover of the algebraic symmetric space
$O(p,q)/[O(1)\times O(p-1,q)]$. But the references also treat analysis on
this cover.)
For completeness, we mention that whenever
\begin{equation}\label{eq:Obigq}
q \ge 1.
\end{equation}
there is another conjugacy class of Cartan subspace, represented by
\begin{equation}\label{eq:splCartanOpq}
{\mathfrak a}_s = \langle e_{1,p+1} + e_{p+1,1} \rangle.
\end{equation}
This one is split, and corresponds to the continuous part of the
Plancherel formula.
The discrete series for the symmetric space $H_{p,q}$ is constructed
as follows. Using the compact Cartan subspace ${\mathfrak a}_c$,
construct a $\theta$-stable parabolic
\begin{equation}\label{eq:qOpq}
{\mathfrak q}^{O(p,q)} = {\mathfrak l}^{O(p,q)} + {\mathfrak u}^{O(p,q)} \subset
{\mathfrak o}(p+q,{\mathbb C});
\end{equation}
the corresponding Levi subgroup is
\begin{equation}
L^{O(p,q)} = [O(p,q)]^{{\mathfrak a}_c} = SO(2) \times O(p-2,q)
\end{equation}
We will need notation for the characters of $SO(2)$:
\begin{equation}
\widehat{SO(2)} = \{\chi_\ell \mid \ell \in {\mathbb Z}\}.
\end{equation}
The discrete series consists of certain irreducible representations
\begin{equation}
A_{{\mathfrak q}^{O(p,q)}}(\lambda), \qquad \lambda\colon L^{O(p,q)} \rightarrow
{\mathbb C}^\times.
\end{equation}
The allowed $\lambda$ are (first) those trivial on
\begin{equation}
L^{O(p,q)} \cap O(p-1,q) = O(p-2,q).
\end{equation}
These are precisely the characters of $SO(2)$, and so are indexed by
integers $\ell \in {\mathbb Z}$. Second, there is a positivity
requirement
\begin{equation}
\ell + (p+q-2)/2 > 0.
\end{equation}
We write
\begin{equation}\label{eq:OAq}\begin{aligned}
\lambda(\ell) &=_{\text{def}} \chi_\ell \otimes 1 \colon L^{O(p,q)}\rightarrow
{\mathbb C}^\times, \\ \pi^{O(p,q)}_\ell &= A_{{\mathfrak
q}^{O(p,q)}}(\lambda_\ell) \qquad
(\ell > (2-p-q)/2).
\end{aligned}\end{equation}
The infinitesimal character of this representation is
\begin{equation}
\text{infl char}(\pi_\ell^{O(p,q)}) = (\ell +
(p+q-2)/2, (p+q-4)/2, (p+q-6)/2,\cdots).
\end{equation}
The discrete part of the Plancherel decomposition is
\begin{equation}\label{eq:Opqdisc}
L^2(H_{p,q})_{\text{disc}} = \sum_{\ell > -(p+q-2)/2}
\pi_\ell^{O(p,q)}.
\end{equation}
This decomposition appears in \cite{StrH}*{page 360}, and
\cite{RossH}*{page 449, Theorem 10, and page 471}. What Strichartz
calls $N$ and $n$ are for us $p$ and $q$; his $d$ is our $\ell$. What
Rossmann calls $q$ and $p$ are for us $p$ and $q$; his $\nu -\rho$ is
our $\ell$; and his $\rho$ is $(p+q-2)/2$. The
identification of the representations as cohomologically induced
may be found in \cite{Virr}*{Theorem 2.9}.
\end{subequations}
\begin{subequations} \label{se:Opqorbit}
Here is the orbit method perspective. Just as for $O(n)$, we use a
trace form to identify ${\mathfrak g}_0^*$ with ${\mathfrak
g}_0$. We find
\begin{equation}
({\mathfrak g}_0/{\mathfrak h}_0)^* \simeq {\mathbb R}^{p-1,q},
\end{equation}
respecting the action of $H=O(p-1,q)$. The orbits of $H$ of largest
dimension are given by the value of the quadratic form: positive for
the orbits represented by nonzero elements $x(e_{12}-e_{21})$ of the
compact Cartan subspace of \eqref{eq:cptCartanOpq}; negative for
nonzero elements of the split Cartan subspace
$y(e_{1,p+1}+ e_{p+1,1})$; and zero for the nilpotent element
$(e_{12}-e_{21}+e_{1,p+1}+e_{p+1,1})$.
Define
\begin{equation}\begin{aligned}
\ell_{\text{orbit}} &= \ell + (n-2)/2,\\
\lambda(\ell_{\text{orbit}}) &=
\ell_{\text{orbit}}\cdot((e_{12}-e_{21})/2).
\end{aligned}
\end{equation}
Then the coadjoint orbits for discrete series have representatives
in the compact Cartan subspace
\begin{equation}
\pi_\ell^{O(p,q)} = \pi(\text{orbit}\ \lambda(\ell_{\text{orbit}})).
\end{equation}
Now this representation is an irreducible unitary cohomologically
induced representation whenever
\begin{equation}
\ell_{\text{orbit}} > 0 \iff \ell > -(n-2)/2.
\end{equation}
One of the advantages of the orbit method picture is that the
condition $\ell_{\text{orbit}} >0$ is simpler than the one
$\ell>(-(n-2)/2)$ arising from more straightforward representation
theory as in \eqref{eq:OAq}. Of course we always need also the
integrality condition
\begin{equation}
\ell_{\text{orbit}} \equiv (n-2)/2 \pmod{\mathbb Z} \iff \ell \equiv 0
\pmod{\mathbb Z}.
\end{equation}
\end{subequations}
\begin{subequations} \label{se:Opqcont}
For completeness we mention also the continuous part of the Plancherel
decomposition. The split Cartan subspace ${\mathfrak a}_s$ (defined
above as long as $p$ and $q$ are each at least $1$) gives rise to a
real parabolic subgroup
\begin{equation}\label{eq:POpq}
P^{O(p,q)} = M^{O(p,q)} A_s N^{O(p,q)}, \qquad M^{O(p,q)} = \{\pm
1\} \times O(p-1,q-1).
\end{equation}
Here $A_s = \exp({\mathfrak a}_s) \simeq {\mathbb R}$, and $\{\pm 1\}$
is
$$O(1)_\Delta \subset O(1)\times O(1) \subset O(1,1);$$
we have
$$\{\pm 1\} \times A_s = SO(1,1) \simeq {\mathbb R}^\times,$$
an algebraic split torus. Therefore
\begin{equation}\label{eq:POpqB}
P^{O(p,q)} = SO(1,1)\times O(p-1,q-1)\times N^{O(p,q)}.
\end{equation}
The characters of $SO(1,1)$ are
\begin{equation}
\widehat{SO(1,1)} = \{\chi_{\epsilon,\nu} \mid \epsilon \in {\mathbb
Z}/2{\mathbb Z}, \nu \in {\mathbb C}\}, \qquad
\chi_{\epsilon,\nu}(r) = |r|^\nu \cdot \sgn(r)^\epsilon.
\end{equation}
We define
\begin{equation}
\pi_{\epsilon,\nu}^{O(p,q)} =
\Ind_{P^{O(p,q)}}^{O(p,q)}\left(\chi_{\epsilon,\nu}\otimes
1_{O(p-1,q-1)}\otimes 1_{N^{O(p,q)}}\right).
\end{equation}
Here (in contrast to the definition of discrete series
$\pi^{O(p,q)}_\ell$) we use normalized induction, with a $\rho$
shift. As a consequence, the infinitesimal character of this
representation is
\begin{equation}
\text{infl char}(\pi_{\epsilon,\nu}^{O(p,q)}) = (\nu, (p+q-4)/2,
(p+q-6)/2,\cdots);
\end{equation}
The continuous part of the Plancherel decomposition is
\begin{equation}\label{eq:Opqcont}
L^2(H_{p,q})_{\text{cont}} = \sum_{\epsilon \in {\mathbb
Z}/2{\mathbb Z}} \int_{\nu\in i{\mathbb R_{\ge 0}}}
\pi_{\epsilon,\nu}^{O(p,q)}.
\end{equation}
Just as for the discrete part of the decomposition, {\em all} (not
just almost all) of the representations $\pi_{\epsilon,\nu}^{O(p,q)}$
are irreducible (always for $\nu \in i{\mathbb R}$).
There is an orbit-theoretic formulation of these parameters as well,
corresponding to elements $-i\nu\cdot(e_{1,p+1} + e_{p+1,1})/2$ of the split
Cartan subspace. We omit the details.
\end{subequations}
\begin{subequations}\label{se:OKbranch}
We will need to understand the restriction of $\pi^{O(p,q)}_\ell$ to
the maximal compact subgroup
\begin{equation}
K = O(p)\times O(q) \subset O(p,q).
\end{equation}
This computation requires knowing
\begin{equation}
L^{O(p,q)} \cap K = SO(2)\times O(p-2)\times O(q), \qquad {\mathfrak
u\cap s} = \chi_{1}\otimes 1 \otimes {\mathbb C}^{q};
\end{equation}
here ${\mathfrak g} = {\mathfrak k}\oplus {\mathfrak s}$ is the
complexified Cartan decomposition. Consequently
\begin{equation}
S^m({\mathfrak u\cap s}) = \chi_m \otimes 1 \otimes S^m({\mathbb
C}^{q}) = \sum_{0\le k \le m/2} \chi_m \otimes 1 \otimes \pi^{O(q)}_{m-2k}.
\end{equation}
Now an analysis of the Blattner formula for restricting
cohomologically induced representations to $K$ gives
\begin{equation}\label{eq:Obranch}
\pi_\ell^{O(p,q)}|_{O(p)\times O(q)} =
\sum_{m=0}^\infty \quad \sum_{0\le k \le m/2}\pi^{O(p)}_{m+\ell + q}
\otimes \pi^{O(q)}_{m-2k}.
\end{equation}
If $p$ is much larger than $q$, then
some of the parameters for representations of $O(p)$ are negative.
Those representations should be understood to be zero.
A description of the restriction to $O(p)\times O(q)$ is in
\cite{RossH}*{Lemma 11}. In Rossmann's coordinates, what is written is
$$ \begin{aligned}
\{ \pi_m^{O(p)} \otimes \pi_n^{O(q)} &\mid -(m+(p-2)/2) +
(n+(q-2)/2) \ge \nu, \\ m+n &\equiv \nu -\rho - p \pmod{2}\}.
\end{aligned}$$
Converting to our coordinates as explained after \eqref{eq:Opqdisc}
gives
\begin{equation}\begin{aligned}
\{ \pi_m^{O(p)} \otimes \pi_n^{O(q)} &\mid (m+(p-2)/2) -
(n+(q-2)/2) \ge \nu, \\
m-n &\equiv \nu -\rho + q \pmod{2}\},
\end{aligned}\end{equation}
or equivalently
\begin{equation}
\{ \pi_m^{O(p)} \otimes \pi_n^{O(q)} \mid m - n \ge \ell + q-1, \quad
m-n \equiv \ell + q \pmod{2}\}.
\end{equation}
The congruence condition makes the inequality into
$$m-n \ge \ell + q,$$
which matches the description in \eqref{eq:Obranch}
Finally, we record the easier formulas
\begin{equation}\label{eq:Ocontbranch}
\pi_{\epsilon,\nu}^{O(p,q)}|_{O(p)\times O(q)} =
\sum_{\substack{m,m' \ge 0\\ m-m' \equiv \epsilon \pmod{2}}} \pi^{O(p)}_m
\otimes \pi^{O(q)}_{m'}.
\end{equation}
\end{subequations}
| 3,958 | 71,259 |
en
|
train
|
0.35.15
|
\section{Hermitian hyperboloids}
\label{sec:Upq}
\setcounter{equation}{0}
\begin{subequations}\label{se:Upq}
In this section we see what the ideas from Sections \ref{sec:invt} and
\ref{sec:Opq} say about the discrete series of the non-symmetric
spherical spaces
\begin{equation}
H_{2p,2q} = \{v\in {\mathbb C}^{p,q} \mid \langle
v,v\rangle_{p,q} = 1\} = U(p,q)/U(p-1,q).
\end{equation}
Here $\langle,\rangle_{p,q}$ is the standard Hermitian form of
signature $(p,q)$ on ${\mathbb C}^{p+q}$. The inclusion of the right side
in the middle is just given by the action of the
unitary group on the basis vector $e_1$; surjectivity is Witt's
theorem for Hermitian forms. These discrete series were
completely described by Kobayashi in \cite{Kob}*{Theorem 6.1}.
To simplify many formulas, we write in this section
\begin{equation}\label{eq:Unpq}
n = p+q.
\end{equation}
Our approach (like Kobayashi's) is to restrict the discrete series
representations $\pi_\ell^{O(2p,2q)}$ of \eqref{eq:Opqdisc} to
$U(p,q)$.
We should mention at this point that the homogeneous space
$U(n)/U(n-1)$ has another noncompact real form $GL(n,{\mathbb
R})/GL(n-1,{\mathbb R})$, arising from the inclusion
\begin{equation}
GL(n,{\mathbb R}) \hookrightarrow O(n,n)
\end{equation}
as a real Levi subgroup. For this real form (as Kobayashi observes)
the discrete series representations $\pi_\ell^{O(n,n)}$ decompose
continuously on restriction to $GL(n,{\mathbb R})$, and consequently
this homogeneous space has no discrete series. (More precisely, the
character $x-y$ of the center of $U(1)$ of $U(p,q)$ (an integer)
appearing in the analysis below must be replaced by a character of the
center ${\mathbb R}^\times$ of $GL(n,{\mathbb R})$ (a real number and
a sign).)
We begin by computing the restriction to $U(p)\times U(q)$.
What is good about this is that the representations of $O(2p)$ and
$O(2q)$ appearing in
\eqref{eq:Obranch} are representations appearing in the action of $O$
on spheres. We already computed (in Theorem \ref{thm:OUcptbranch}) how
those branch to unitary groups. The conclusion is
\begin{small}\begin{equation}\label{e:OUbranch}
\pi^{O(2p,2q)}_{_\ell}|_{U(p)\times U(q)} =
\sum_{\substack{0\le b,c \\[.2ex] b+c \ge \ell+2q}} \quad
\sum_{\substack{0\le b',c' \\[.2ex] b'+c' \le b+c -\ell-2q
\\[.2ex] b'+c'
\equiv b+c-\ell \pmod{2}}}
\pi^{U(p)}_{b,c}\otimes \pi^{U(q)}_{b',c'}.
\end{equation}\end{small}
\end{subequations}
This calculation, together with Corollary \ref{cor:Cdiff}, proves most of
\begin{proposition}\label{prop:OUbranch} Suppose $p$ and $q$ are
nonnegative integers, each at least two; and suppose $\ell >
-(n-1)$. Then the restriction of the discrete series representation
$\pi^{O(2p,2q)}_\ell$ to $U(p,q)$ is the direct sum of
the one-parameter family of representations
$$\pi^{U(p,q)}_{x,y},\quad
x,y \in {\mathbb Z}, \quad x+y = \ell.$$
The infinitesimal character of $\pi^{U(p,q)}_{x,y}$ corresponds to
the weight
$$(x+(n-1)/2,(n-3)/2,\ldots,-(n-3)/2, -y-(n-1)/2).$$
Restriction to the maximal compact subgroup is
$$\pi^{U(p,q)}_{x,y}|_{U(p)\times U(q)} = \sum_{r,s\ge
0}\sum_{k=0}^{\min(r,s)} \pi^{U(p)}_{x+q+r,y+q+s}\otimes
\pi^{U(q)}_{s-k,r-k}.$$
If one of the two subscripts in a $U(p)$ representation is negative,
that term is to be interpreted as zero.
Each of the representations $\pi^{U(p,q)}_{x,y}$ is irreducible.
\end{proposition}
The ``one parameter'' referred to in the proposition is $x-y$; the
pair $(x,y)$ can be thought of as a single parameter because of the
constraint $x+y=\ell$. What we have done is sorted the representations
of $U(p)\times U(q)$ appearing in \eqref{e:OUbranch} according to the
character of the center $U(1)$ of $U(p,q)$; this character is
$(b-c)+(b'-c')$, and we call it $x-y$ in the rearrangement in
Proposition \ref{prop:OUbranch}. The corresponding representation of
$U(p,q)$ (the part of $\pi^{O(2p,2q)}_\ell$ where $U(1)$ acts by
$x-y$) is what we call $\pi^{U(p,q)}_{x,y}$. In order to prove most of
the proposition, we just need to check that the same representations
of $U(p)\times U(q)$ appear in \eqref{e:OUbranch} and in Proposition
\ref{prop:OUbranch}, and this is easy. We will prove the
irreducibility assertion (using \cite{Kob}) after \eqref{e:Aq-smallx}
below.
\begin{subequations}\label{se:Upqrep}
Having identified the restriction to $U(p)\times U(q)$, we record
for completeness Kobayashi's identification of the actual representations of
$U(p,q)$. These come in three families, according to the values of
the integers $x$ and $y$. The families are cohomologically induced
from three $\theta$-stable parabolic subalgebras:
\begin{equation}
{\mathfrak q}^{U(p,q)}_+ = {\mathfrak l}^{U(p,q)}_+ + {\mathfrak
u}^{U(p,q)}_+ \subset {\mathfrak u}(n,{\mathbb C});
\end{equation}
with Levi subgroup
\begin{equation}
L^{U(p,q)}_+ = U(1)_p\times U(1)_q \times U(p-1,q-1);
\end{equation}
\begin{equation}
{\mathfrak q}^{U(p,q)}_0 = {\mathfrak l}^{U(p,q)}_0 + {\mathfrak
u}^{U(p,q)}_0 \subset {\mathfrak u}(n,{\mathbb C});
\end{equation}
with Levi subgroup
\begin{equation}
L^{U(p,q)}_0 = U(1)_p \times U(p-2,q)\times U(1)_p;
\end{equation}
and
\begin{equation}
{\mathfrak q}^{U(p,q)}_- = {\mathfrak l}^{U(p,q)}_- + {\mathfrak
u}^{U(p,q)}_- \subset {\mathfrak u}(n,{\mathbb C});
\end{equation}
with Levi subgroup
\begin{equation}
L^{U(p,q)}_- = U(p-1,q-1)\times U(1)_q \times U(1)_p.
\end{equation}
(We write $U(1)_p$ for a coordinate $U(1) \subset U(p)$, and $U(1)_q
\subset U(q)$ similarly. More complete descriptions of these
parabolics are in \cite{Kob}.)
Suppose first that
\begin{equation}\label{eq:bigx}
x> \ell + (n-1)/2, \qquad y < -(n-1)/2.
\end{equation}
(Since $x+y=\ell$, these two inequalities are equivalent.) Write
$\xi_x$ for the character of $U(1)$ corresponding to $x\in {\mathbb Z}$.
Consider the one-dimensional character
\begin{equation}
\lambda^+_{x,y} = \xi_x\otimes \xi_{-(y + n-2)} \otimes {\det}^1
\end{equation}
of $L^{U(p,q)}_+$. What Kobayashi proves in \cite{Kob}*{Theorem 6.1}
is
\begin{equation}
\pi^{U(p,q)}_{x,y} = A_{{\mathfrak q}^{U(p,q)}_+}(\lambda^+_{x,y})
\qquad (x > \ell + (n-1)/2).
\end{equation}
Suppose next that
\begin{equation}\label{eq:middlex}
\ell + (n-1)/2 \ge x \ge -(n-1)/2, \qquad -(n-1)/2\le y \le
\ell + (n-1)/2.
\end{equation}
(Since $x+y=\ell$, these two pairs of inequalities are equivalent.)
Consider the one-dimensional character
\begin{equation}
\lambda^0_{x,y} = \xi_x\otimes 1\otimes \xi_{-y}
\end{equation}
of $L^{U(p,q)}_0$. Kobayashi's result in \cite{Kob}*{Theorem 6.1}
is now
\begin{equation}
\pi^{U(p,q)}_{x,y} = A_{{\mathfrak q}^{U(p,q)}_0}(\lambda^0_{x,y})
\qquad (-(n-1)/2 \le x \le \ell + (n-1)/2).
\end{equation}
The remaining case is
\begin{equation}\label{eq:smallx}
x < -(n-1)/2, \qquad y > \ell + (n-1)/2.
\end{equation}
(Since $x+y=\ell$, these two inequalities are equivalent.) Write
\begin{equation}
\lambda^-_{x,y} = {\det}^{-1}\otimes \xi_{x+n-2}\otimes \xi_{-y}
\end{equation}
of $L^{U(p,q)}_-$. In this case Kobayashi proves
\begin{equation}\label{e:Aq-smallx}
\pi^{U(p,q)}_{x,y} = A_{{\mathfrak q}^{U(p,q)}_-}(\lambda^-_{x,y})
\qquad (x < -(n-1)/2).
\end{equation}
\end{subequations}
\begin{subequations} \label{se:Upqorbit}
Here is the orbit method perspective. Just as for $U(n)$, we use a
trace form to identify ${\mathfrak g}_0^*$ with ${\mathfrak
g}_0$. The linear functionals vanishing on ${\mathfrak h}_0^*$ are
\begin{equation}
\lambda(t,u,v) = \begin{pmatrix}it & \hskip -2ex u& \hskip -2ex
v\\[.5ex] -\overline u &
\\ \overline v & \text{\Large \hskip 3ex $0_{(n-1)\times (n-1)}$ \hskip
-2ex}\\ \end{pmatrix} \simeq
{\mathbb R} + {\mathbb C}^{p-1,q}
\end{equation}
with $t\in {\mathbb R}$, $u \in {\mathbb C}^{p-1}$, $v\in {\mathbb C}^q$.
The orbits of $H=U(p-1,q)$ of largest dimension are given by the real
number $t$, and the value of the Hermitian form on the vector $(u,v)$:
positive for the orbits represented by nonzero elements
$r(e_{12}-e_{21})$ (nonzero eigenvalues $i(t\pm a)/2$, with $a=(t^2 +
4r^2)^{1/2}$); negative for nonzero elements $s(e_{1,p+1}+ e_{p+1,1})$
(nonzero eigenvalues $i(t\pm a)/2$, with $a=(t^2 - 4s^2)^{1/2}$); and
zero for the nilpotent element $(e_{12}-e_{21}+e_{1,p+1}+e_{p+1,1})$
(two nonzero eigenvalues $it/2$).
Define
\begin{equation}
\ell_{\text{orbit}} = \ell + (n-1), \quad x_{\text{orbit}} = x +
(n-1)/2, \quad y_{\text{orbit}} = y + (n-1)/2.
\end{equation}
The coadjoint orbits for discrete series have representatives
\begin{equation}
\lambda(x_{\text{orbit}},y_{\text{orbit}}) = \begin{cases} ix_{\text{orbit}}e_1
- iy_{\text{orbit}}e_{p+1} & x_{\text{orbit}} > 0 >
y_{\text{orbit}}\\
ix_{\text{orbit}}e_1 + (e_{2,p}-e_{p,2}) & \\
\quad+ (e_{2,p+1}+e_{p+1,2}) & x_{\text{orbit}} > 0 = y_{\text{orbit}}\\
ix_{\text{orbit}}e_1 - iy_{\text{orbit}}e_{p} & x_{\text{orbit}} >
y_{\text{orbit}} > 0\\
iy_{\text{orbit}}e_p + (e_{1,2}-e_{2,1})&\\
\quad +e_{1,p+1}+e_{p+1,1}) & x_{\text{orbit}} = 0 > y_{\text{orbit}}\\
ix_{\text{orbit}}e_p - iy_{\text{orbit}}e_{p+q} & 0 > x_{\text{orbit}} >
y_{\text{orbit}}.
\end{cases}
\end{equation}
Then
\begin{equation}
\pi_{x,y}^{U(p,q)} = \pi(\text{orbit}\ \lambda(x_{\text{orbit}},y_{\text{orbit}})).
\end{equation}
(We have not discussed attaching representations to partly nilpotent
coadjoint orbits like $\lambda(x_{\text{orbit}},0)$ (with
$x_{\text{orbit}} >0$); suffice it to say that the definitions given
above using ${\mathfrak q}_0$ are reasonable ones. It would be equally
reasonable to use instead ${\mathfrak q}_+$. We will see in
\eqref{eq:Aq-big-middle-coincide} that this leads to the same
representation.)
| 3,863 | 71,259 |
en
|
train
|
0.35.16
|
\begin{subequations} \label{se:Upqorbit}
Here is the orbit method perspective. Just as for $U(n)$, we use a
trace form to identify ${\mathfrak g}_0^*$ with ${\mathfrak
g}_0$. The linear functionals vanishing on ${\mathfrak h}_0^*$ are
\begin{equation}
\lambda(t,u,v) = \begin{pmatrix}it & \hskip -2ex u& \hskip -2ex
v\\[.5ex] -\overline u &
\\ \overline v & \text{\Large \hskip 3ex $0_{(n-1)\times (n-1)}$ \hskip
-2ex}\\ \end{pmatrix} \simeq
{\mathbb R} + {\mathbb C}^{p-1,q}
\end{equation}
with $t\in {\mathbb R}$, $u \in {\mathbb C}^{p-1}$, $v\in {\mathbb C}^q$.
The orbits of $H=U(p-1,q)$ of largest dimension are given by the real
number $t$, and the value of the Hermitian form on the vector $(u,v)$:
positive for the orbits represented by nonzero elements
$r(e_{12}-e_{21})$ (nonzero eigenvalues $i(t\pm a)/2$, with $a=(t^2 +
4r^2)^{1/2}$); negative for nonzero elements $s(e_{1,p+1}+ e_{p+1,1})$
(nonzero eigenvalues $i(t\pm a)/2$, with $a=(t^2 - 4s^2)^{1/2}$); and
zero for the nilpotent element $(e_{12}-e_{21}+e_{1,p+1}+e_{p+1,1})$
(two nonzero eigenvalues $it/2$).
Define
\begin{equation}
\ell_{\text{orbit}} = \ell + (n-1), \quad x_{\text{orbit}} = x +
(n-1)/2, \quad y_{\text{orbit}} = y + (n-1)/2.
\end{equation}
The coadjoint orbits for discrete series have representatives
\begin{equation}
\lambda(x_{\text{orbit}},y_{\text{orbit}}) = \begin{cases} ix_{\text{orbit}}e_1
- iy_{\text{orbit}}e_{p+1} & x_{\text{orbit}} > 0 >
y_{\text{orbit}}\\
ix_{\text{orbit}}e_1 + (e_{2,p}-e_{p,2}) & \\
\quad+ (e_{2,p+1}+e_{p+1,2}) & x_{\text{orbit}} > 0 = y_{\text{orbit}}\\
ix_{\text{orbit}}e_1 - iy_{\text{orbit}}e_{p} & x_{\text{orbit}} >
y_{\text{orbit}} > 0\\
iy_{\text{orbit}}e_p + (e_{1,2}-e_{2,1})&\\
\quad +e_{1,p+1}+e_{p+1,1}) & x_{\text{orbit}} = 0 > y_{\text{orbit}}\\
ix_{\text{orbit}}e_p - iy_{\text{orbit}}e_{p+q} & 0 > x_{\text{orbit}} >
y_{\text{orbit}}.
\end{cases}
\end{equation}
Then
\begin{equation}
\pi_{x,y}^{U(p,q)} = \pi(\text{orbit}\ \lambda(x_{\text{orbit}},y_{\text{orbit}})).
\end{equation}
(We have not discussed attaching representations to partly nilpotent
coadjoint orbits like $\lambda(x_{\text{orbit}},0)$ (with
$x_{\text{orbit}} >0$); suffice it to say that the definitions given
above using ${\mathfrak q}_0$ are reasonable ones. It would be equally
reasonable to use instead ${\mathfrak q}_+$. We will see in
\eqref{eq:Aq-big-middle-coincide} that this leads to the same
representation.)
In the orbit method picture the condition \eqref{eq:bigx} simplifies to
\begin{equation}\label{eq:bigxorbit}
x_{\text{orbit}} > y_{\text{orbit}} > 0.
\end{equation}
Similarly, \eqref{eq:smallx} becomes
\begin{equation}\label{eq:smallxorbit}
x_{\text{orbit}} < y_{\text{orbit}}<0.
\end{equation}
Finally, \eqref{eq:middlex} is
\begin{equation}\label{eq:middlexorbit}
x_{\text{orbit}} \ge 0 \ge y_{\text{orbit}};
\end{equation}
equality in either of these inequalities is the case of partially
nilpotent coadjoint orbits.
In all cases we need also the genericity condition
\begin{equation}
\ell_{\text{orbit}} >0 \iff \ell > -(n-1),
\end{equation}
and the integrality conditions
\begin{equation}
x_{\text{orbit}} \equiv (n-1)/2 \pmod{\mathbb Z}, y_{\text{orbit}}
\equiv (n-1)/2 \pmod{\mathbb Z}.
\end{equation}
\end{subequations}
\begin{subequations}
Here now is a sketch of a proof of the irreducibility assertion from
Proposition \ref{prop:OUbranch}. Each of the cohomologically induced
representations above is in the
weakly fair range. The general results for the weakly fair range of
\cite{Vunit} together with \cite{VGLn}*{Section 16} apply to show that
they are irreducible or zero. The key point is that the moment map
for the cotangent bundle to a relevant partial flag variety is
birational onto its image. This is automatic in type $A$, which is why
the arguments in \cite{VGLn} for $GL(n,\mathbb{R})$ also apply to
$U(p,q)$.
We close with a comment about how the three series of derived functor
modules fit together. If we relax the strict inequalities on $x$ (and
$y$) in \eqref{eq:bigx}, then we are at one edge of the weak
inequalities in \eqref{eq:middlex}. For these values of $x$ and $y$
(which occur only when $n$ is odd), namely
\[
(x,y) = \left ( \ell + (n-1)/2, -(n-1)/2 \right),
\]
or equivalently
\[
(x_{\text{orbit}},y_{\text{orbit}}) = \left(
\ell_{\text{orbit}},0\right),
\]
we claim
\begin{equation}\label{eq:Aq-big-middle-coincide}
A_{{\mathfrak q}^{U(p,q)}_+}(\lambda^+_{x,y}) \simeq A_{{\mathfrak
q}^{U(p,q)}_0}(\lambda^0_{x,y}).
\end{equation}
To see this, one can begin by checking they have the same associated
variety: the $U(p,{\mathbb C}) \times U(q,{\mathbb C})$ saturations of
${\mathfrak u}^{U(p,q)}_+ \cap \mathfrak{s}$ and ${\mathfrak
u}^{U(p,q)}_0 \cap \mathfrak{s}$ coincide. (The dense orbit of
$U(p,{\mathbb C}) \times U(q,{\mathbb C})$ is one of the two
possibilities with one Jordan block of size $3$ and the others of size
$1$.) A little further checking shows that they also have the same
annihilator: for $\ell \leq (n-2)/2$, given the associated variety
calculation, there is a unique possibility for the annihilator; a
slightly more refined analysis handles larger $\ell$. Given that
their annihilators and associated varieties are the same, the main
result of \cite{BVUpq} implies \eqref{eq:Aq-big-middle-coincide}.
Similarly, for the other edge of the inequalities in
\eqref{eq:middlex}, namely
\[
(x,y) = \left (-(n-1)/2, \ell+(n-1)/2 \right),
\]
we have
\begin{equation}\label{eq:Aq-small-middle-coincide}
A_{{\mathfrak q}^{U(p,q)}_0}(\lambda^0_{x,y}) \simeq A_{{\mathfrak
q}^{U(p,q)}_-}(\lambda^-_{x,y})
\end{equation}
by a similar argument.
\end{subequations}
| 2,189 | 71,259 |
en
|
train
|
0.35.17
|
\section{Quaternionic hyperboloids}
\label{sec:Sppq}
\setcounter{equation}{0}
\begin{subequations}\label{se:Sppq}
In this section we use the ideas from Section \ref{sec:invt} to investigate the
discrete series of the non-symmetric spherical spaces
\begin{equation}\begin{aligned}
H_{4p,4q} &= \{v\in {\mathbb H}^{p,q} \mid \langle
v,v\rangle_{p,q} = 1\}\\ &= [Sp(p,q)\times Sp(1)]/[Sp(p-1,q)\times
Sp(1)_\Delta].
\end{aligned}\end{equation}
Here $\langle,\rangle_{p,q}$ is the standard Hermitian form of
signature $(p,q)$ on ${\mathbb H}^{p+q}$. We are using the action of
a real form of the enlarged group from \eqref{eq:Spbig}, namely
\begin{equation}
Sp(p,q) \times Sp(1) = Sp(p,q)_{\text{linear}} \times
Sp(1)_{\text{scalar}};
\end{equation}
The inclusion of the last
side of the equality (for $H_{4p,4q}$) in the middle is just given by
the action of this enlarged quaternionic unitary group on the basis
vector $e_1$; surjectivity is Witt's theorem for quaternionic
Hermitian forms. To avoid talking about degenerate cases, we will
assume
\begin{equation}
p, q \ge 2.
\end{equation}
Just as in Section \ref{sec:Upq}, we will simplify many formulas by
writing
\begin{equation}\label{eq:Spnpq}
n = p+q.
\end{equation}
The homogeneous space
$Sp(n)/Sp(n-1)$ has another noncompact real form $[Sp(2n,{\mathbb
R})\times Sp(2,{\mathbb R})]/[Sp(2(n-1),{\mathbb R})\times
Sp(2,{\mathbb R})_\Delta]$, arising from an inclusion
\begin{equation}
Sp(2n,{\mathbb R})\times Sp(2,{\mathbb R})\ \hookrightarrow O(2n,2n).
\end{equation}
This real form certainly has discrete series: we expect that the
discrete summands of the restriction of $\pi_\ell^{O(2n,2n)}$ are
indexed by discrete series representations of $Sp(2,{\mathbb R})$,
just as we find below (for $Sp(p,q)$) that they are indexed by
irreducible representations of the compact group $Sp(1)$. But we have
not carried out this analysis.
Our goal is to restrict the discrete series representations
$\pi^{O(4p,4q)}_\ell$ of \eqref{eq:Opqdisc} to $Sp(p,q)$, and so to
understand some representations in the discrete series of $(Sp(p,q)\times
Sp(1))/(Sp(p-1,q)\times Sp(1))$.
\end{subequations}
\begin{subequations}\label{se:OSpbranch}
We have calculated in Theorem \ref{thm:OSpcptbranch} how the $O(4p)$
and $O(4q)$ representations appearing in \eqref{eq:Obranch} restrict
to $Sp$. The result is
\begin{small}\begin{equation}\label{e:OSpBigBranch}
\pi_\ell^{O(4p,4q)}|_{[Sp(p)\times Sp(1)]\times [Sp(q)\times Sp(1)]} =
\sum_{\substack {m=0\\[.1ex] 0 \le k \le m/2}}^\infty \
\sum_{\substack{0\le e \le d\\[.1ex] 0\le e'\le d'\\[.2ex] d+e =
m+\ell+4q\\[.1ex] d'+e' = m-2k}}
\pi^{Sp(p)\times Sp(1)}_{d,e}\otimes \pi^{Sp(q)\times Sp(1)}_{d',e'}.
\end{equation}\end{small}
The group to which we are restricting here is actually a little larger
than the maximal compact subgroup of $Sp(p,q)\times Sp(1)$, which is
\begin{equation}
Sp(p) \times Sp(q) \times Sp(1)_\Delta;
\end{equation}
the subscript $\Delta$ indicates that this $Sp(1)$ factor
(corresponding to scalar multiplication on ${\mathbb H}^{p,q})$) is
diagonal in the $Sp(1)\times Sp(1)$ of \eqref{e:OSpBigBranch}
(corresponding to separate scalar multiplications on ${\mathbb H}^p$
and ${\mathbb H}^q$).
The branching $(G\times G)|_{G_\Delta}$ is tensor product
decomposition, which is very simple for $Sp(1)$. We find
\begin{small}\begin{equation}\label{e:OSpcptBranch}\begin{aligned}
&\pi_\ell^{O(4p,4q)}|_{[Sp(p)\times Sp(q)\times Sp(1)]} =\\
&\sum_{\substack {m=0\\[.2ex] 0 \le k \le m/2}}^\infty \
\sum_{\substack{0\le e \le d\\[.1ex] 0\le e'\le d'\\[.2ex] d+e =
m+\ell+4q\\[.1ex] d'+e' = m-2k}}
\sum_{j=0}^{\min(d-e,d'-e')}
\pi^{Sp(p)}_{d,e}\otimes \pi^{Sp(q)}_{d',e'}\otimes \pi^{Sp(1)}_{d+d'-e-e'-2j}.
\end{aligned}\end{equation}\end{small}
It will be useful to rewrite this formula. The indices $m$ and $k$
serve only to bound some of the other indices, so we can eliminate
them by rewriting the bounds. We find
\begin{small}\begin{equation}\label{e:OSpcptBranch2}\begin{aligned}
&\pi_\ell^{O(4p,4q)}|_{[Sp(p)\times Sp(q)\times Sp(1)]} =\\[.5ex]
&\sum_{\substack{0\le e \le d\quad 0\le e'\le d'\\[.4ex] d'+e' \le d+e-\ell-4q
\\[.4ex] d'+e' \equiv d+e - \ell \pmod{2}}}
\sum_{\substack{|(d-e)-(d'-e')|\le f \\[.2ex] \quad \le
(d-e)+(d'-e')\\[.4ex] \quad f\equiv
(d-e)+(d'-e')\pmod{2}}}
\pi^{Sp(p)}_{d,e}\otimes \pi^{Sp(q)}_{d',e'}\otimes \pi^{Sp(1)}_f.
\end{aligned}\end{equation}\end{small}
For each of these representations of $K$, define integers $x$ and $y$
by solving the equations
\begin{equation}
x+y = \ell, \qquad x-y = f.
\end{equation}
The congruence condition on $f$ guarantees that $x$ and $y$ are
indeed integers. Conversely, given any integers $x$ and $y$ satisfying
\begin{equation}
x+y=\ell, \qquad x\ge y
\end{equation}
we can define
\begin{equation}
\begin{split}
\pi^{Sp(p,q)\times Sp(1)}_{x,y} = \text{subrepresentation of
$\pi^{O(4p,4q)}_\ell|_{Sp(p,q)\times Sp(1)}$}\\
\; \text{where $Sp(1)$ acts
with infl. char. $x-y+1$.}
\end{split}
\end{equation}
Equivalently, we are asking that $Sp(1)$ act by a multiple of
$\pi^{Sp(1)}_{x-y}$.
\end{subequations}
This calculation, together with Corollary \ref{cor:Hdiff}, proves most of
\begin{proposition}\label{prop:OSpbranch} Suppose $p$ and $q$ are
nonnegative integers, each at least two; and suppose $\ell > -2n
+1$. Then the restriction of the discrete series representation
$\pi^{O(4p,4q)}_\ell$ to $Sp(p,q)\times Sp(1)$ is the direct sum of
the one-parameter family of representations
$$\pi^{Sp(p,q)\times Sp(1)}_{x,y},\quad
x\ge y \in {\mathbb Z}, \quad x+y =
\ell.$$
The infinitesimal character of $\pi^{Sp(p,q)\times Sp(1)}_{x,y}$ corresponds to
the weight
$$(x+n, y+n-1,n-2,\ldots,1)(x-y+1).$$
Restriction to the maximal compact subgroup is
\begin{small}\begin{equation*}\begin{aligned}
& \pi^{Sp(p,q)\times Sp(1)}_{x,y}|_{Sp(p)\times Sp(q) \times Sp(1)} =\\[.5ex]
&\sum_{\substack{0\le e \le d\quad 0\le e'\le d'\\[.4ex] d'+e' \le d+e-(x+y)-4q
\\[.4ex] d'+e' \equiv d+e - (x+y) \pmod{2}}}
\sum_{\substack{|(d-e)-(d'-e')|\le x-y \\[.2ex] x-y \le
(d-e)+(d'-e')\\[.4ex] \quad
(d-e)+(d'-e') \equiv x-y \pmod{2}}}
\pi^{Sp(p)}_{d,e}\otimes \pi^{Sp(q)}_{d',e'}\otimes \pi^{Sp(1)}_{x-y}.
\end{aligned}\end{equation*}\end{small}
Each of the representations $\pi^{Sp(p,q)}_{x,y}$ is irreducible.
\end{proposition}
We will prove the irreducibility assertions (using \cite{Kob}) after
\eqref{eq:Aq-Spsmallx} below.
\begin{subequations}\label{se:Sppqrep}
Having identified the restriction to $Sp(p)\times Sp(q)\times Sp(1)$,
we want to record Kobayashi's identification of the actual
representations of $Sp(p,q)\times Sp(1)$. These come in two
families, according to
the values of the integers $x$ and $y$. The families are
cohomologically induced from two $\theta$-stable parabolic
subalgebras. The first is
\begin{equation}
{\mathfrak q}_+^{Sp(p,q)\times Sp(1)} = {\mathfrak l}_+^{Sp(p,q)\times
Sp(1)} + {\mathfrak u}_+^{Sp(p,q)\times Sp(1)} \subset
{\mathfrak s}{\mathfrak p}(n,{\mathbb C}) \times {\mathfrak
s}{\mathfrak p}(1,{\mathbb C}),
\end{equation}
with Levi subgroup
\begin{equation}
L_+^{Sp(p,q)\times Sp(1)} = [U(1)_p\times U(1)_q\times Sp(p-1,q-1)] \times U(1).
\end{equation}
(The first three factors are in $Sp(p,q)$. We write $U(1)_p$ for a
coordinate $U(1) \subset U(p)$, and $U(1)_q \subset U(q)$ similarly.)
The second parabolic is
\begin{equation}
{\mathfrak q}_0^{Sp(p,q)\times Sp(1)} = {\mathfrak l}_0^{Sp(p,q)\times
Sp(1)} + {\mathfrak u}_0^{Sp(p,q)\times Sp(1)} \subset
{\mathfrak s}{\mathfrak p}(n,{\mathbb C}) \times {\mathfrak
s}{\mathfrak p}(1,{\mathbb C}),
\end{equation}
with Levi subgroup
\begin{equation}
L_0^{Sp(p,q)\times Sp(1)} = [U(1)_p\times U(1)_p\times Sp(p-2,q)] \times U(1).
\end{equation}
More complete descriptions of these parabolics are in \cite{Kob}.)
Suppose first that
\begin{equation}\label{eq:Spbigx}
x> \ell + (n-1), \qquad y < -(n-1).
\end{equation}
(Since $x+y=\ell$, these two inequalities are equivalent.) Write
$\xi_x$ for the character of $U(1)$ corresponding to $x\in {\mathbb Z}$.
Consider the one-dimensional character
\begin{equation}\label{eq:lam-Spbigx}
\lambda^+_{x,y} = \left[\xi_x\otimes \xi_{-(y+2n-2)} \otimes
1\right]\otimes \xi_{x-y}
\end{equation}
of $L^{Sp(p,q)\times Sp(1)}_+$. What Kobayashi proves in \cite{Kob}*{Theorem 6.1}
is
\begin{equation}\label{eq:Aq-Spbigx}
\pi^{Sp(p,q)\times Sp(1)}_{x,y} = A_{{\mathfrak q}^{Sp(p,q)\times
Sp(1)}_+}(\lambda^+_{x,y}) \qquad x > \ell + (n-1).
\end{equation}
Suppose next that
\begin{equation}\label{eq:Spsmallx}
\ell + (n-1) \ge x > \ell/2, \qquad -(n-1)\le y < \ell/2.
\end{equation}
(Since $x+y=\ell$, these two pairs of inequalities are equivalent.)
Consider the one-dimensional character
\begin{equation}\label{eq:lam-Spsmallx}
\lambda^0_{x,y} = \left[\xi_x\otimes \xi_y\otimes 1\right]\otimes \xi_{x-y}
\end{equation}
of $L^{Sp(p,q)\times Sp(1)}_0$. Kobayashi's result in \cite{Kob}*{Theorem 6.1}
is now
\begin{equation}\label{eq:Aq-Spsmallx}
\pi^{Sp(p,q)\times Sp(1)}_{x,y} = A_{{\mathfrak q}^{Sp(p,q)\times
Sp(1)}_0}(\lambda^0_{x,y})
\qquad \ell/2 < x \le \ell + (n-1)).
\end{equation}
\end{subequations}
\begin{subequations}\label{se:Sppqorbit}
Here is the orbit method perspective. Use a
trace form to identify ${\mathfrak g}_0^*$ with ${\mathfrak
g}_0$. Linear functionals vanishing on ${\mathfrak h}_0^*$ are
quaternionic matrices
\begin{equation}
\lambda(z,u,v) = \left[\begin{pmatrix}z & \hskip -2ex u& \hskip -2ex
v\\[.5ex] -\overline u &
\\ \overline v & \text{\Large \hskip 3ex $0_{(n-1)\times (n-1)}$ \hskip
-2ex}\\ \end{pmatrix},-z\right] \simeq
{\mathfrak s}{\mathfrak p}(1) + {\mathbb H}^{p-1,q}
\end{equation}
with $z\in {\mathfrak s}{\mathfrak p}(1)$ (the purely imaginary
quaternions), $u \in {\mathbb H}^{p-1}$, $v\in {\mathbb H}^q$.
| 3,997 | 71,259 |
en
|
train
|
0.35.18
|
Each of the representations $\pi^{Sp(p,q)}_{x,y}$ is irreducible.
\end{proposition}
We will prove the irreducibility assertions (using \cite{Kob}) after
\eqref{eq:Aq-Spsmallx} below.
\begin{subequations}\label{se:Sppqrep}
Having identified the restriction to $Sp(p)\times Sp(q)\times Sp(1)$,
we want to record Kobayashi's identification of the actual
representations of $Sp(p,q)\times Sp(1)$. These come in two
families, according to
the values of the integers $x$ and $y$. The families are
cohomologically induced from two $\theta$-stable parabolic
subalgebras. The first is
\begin{equation}
{\mathfrak q}_+^{Sp(p,q)\times Sp(1)} = {\mathfrak l}_+^{Sp(p,q)\times
Sp(1)} + {\mathfrak u}_+^{Sp(p,q)\times Sp(1)} \subset
{\mathfrak s}{\mathfrak p}(n,{\mathbb C}) \times {\mathfrak
s}{\mathfrak p}(1,{\mathbb C}),
\end{equation}
with Levi subgroup
\begin{equation}
L_+^{Sp(p,q)\times Sp(1)} = [U(1)_p\times U(1)_q\times Sp(p-1,q-1)] \times U(1).
\end{equation}
(The first three factors are in $Sp(p,q)$. We write $U(1)_p$ for a
coordinate $U(1) \subset U(p)$, and $U(1)_q \subset U(q)$ similarly.)
The second parabolic is
\begin{equation}
{\mathfrak q}_0^{Sp(p,q)\times Sp(1)} = {\mathfrak l}_0^{Sp(p,q)\times
Sp(1)} + {\mathfrak u}_0^{Sp(p,q)\times Sp(1)} \subset
{\mathfrak s}{\mathfrak p}(n,{\mathbb C}) \times {\mathfrak
s}{\mathfrak p}(1,{\mathbb C}),
\end{equation}
with Levi subgroup
\begin{equation}
L_0^{Sp(p,q)\times Sp(1)} = [U(1)_p\times U(1)_p\times Sp(p-2,q)] \times U(1).
\end{equation}
More complete descriptions of these parabolics are in \cite{Kob}.)
Suppose first that
\begin{equation}\label{eq:Spbigx}
x> \ell + (n-1), \qquad y < -(n-1).
\end{equation}
(Since $x+y=\ell$, these two inequalities are equivalent.) Write
$\xi_x$ for the character of $U(1)$ corresponding to $x\in {\mathbb Z}$.
Consider the one-dimensional character
\begin{equation}\label{eq:lam-Spbigx}
\lambda^+_{x,y} = \left[\xi_x\otimes \xi_{-(y+2n-2)} \otimes
1\right]\otimes \xi_{x-y}
\end{equation}
of $L^{Sp(p,q)\times Sp(1)}_+$. What Kobayashi proves in \cite{Kob}*{Theorem 6.1}
is
\begin{equation}\label{eq:Aq-Spbigx}
\pi^{Sp(p,q)\times Sp(1)}_{x,y} = A_{{\mathfrak q}^{Sp(p,q)\times
Sp(1)}_+}(\lambda^+_{x,y}) \qquad x > \ell + (n-1).
\end{equation}
Suppose next that
\begin{equation}\label{eq:Spsmallx}
\ell + (n-1) \ge x > \ell/2, \qquad -(n-1)\le y < \ell/2.
\end{equation}
(Since $x+y=\ell$, these two pairs of inequalities are equivalent.)
Consider the one-dimensional character
\begin{equation}\label{eq:lam-Spsmallx}
\lambda^0_{x,y} = \left[\xi_x\otimes \xi_y\otimes 1\right]\otimes \xi_{x-y}
\end{equation}
of $L^{Sp(p,q)\times Sp(1)}_0$. Kobayashi's result in \cite{Kob}*{Theorem 6.1}
is now
\begin{equation}\label{eq:Aq-Spsmallx}
\pi^{Sp(p,q)\times Sp(1)}_{x,y} = A_{{\mathfrak q}^{Sp(p,q)\times
Sp(1)}_0}(\lambda^0_{x,y})
\qquad \ell/2 < x \le \ell + (n-1)).
\end{equation}
\end{subequations}
\begin{subequations}\label{se:Sppqorbit}
Here is the orbit method perspective. Use a
trace form to identify ${\mathfrak g}_0^*$ with ${\mathfrak
g}_0$. Linear functionals vanishing on ${\mathfrak h}_0^*$ are
quaternionic matrices
\begin{equation}
\lambda(z,u,v) = \left[\begin{pmatrix}z & \hskip -2ex u& \hskip -2ex
v\\[.5ex] -\overline u &
\\ \overline v & \text{\Large \hskip 3ex $0_{(n-1)\times (n-1)}$ \hskip
-2ex}\\ \end{pmatrix},-z\right] \simeq
{\mathfrak s}{\mathfrak p}(1) + {\mathbb H}^{p-1,q}
\end{equation}
with $z\in {\mathfrak s}{\mathfrak p}(1)$ (the purely imaginary
quaternions), $u \in {\mathbb H}^{p-1}$, $v\in {\mathbb H}^q$.
The orbits of $H=Sp(p-1,q)\times Sp(1)_\Delta$ of largest dimension are given
by $|z|$, and the value of the Hermitian form on the vector $(u,v)$:
positive for the orbits represented by nonzero elements
$r(e_{12}-e_{21})$ (nonzero eigenvalues $i(|z|\pm a)/2$, with $a=(|z|^2 +
4r^2)^{1/2}$); negative for nonzero elements $s(e_{1,p+1}+ e_{p+1,1})$
(nonzero eigenvalues $i(|z|\pm a)/2$, with $a=(|z|^2 - 4s^2)^{1/2}$); and
zero for the nilpotent element $(e_{12}-e_{21}+e_{1,p+1}+e_{p+1,1})$
(nonzero eigenvalues $i|z|/2$).
Define
\begin{equation}
\ell_{\text{orbit}} = \ell + (2n-1), \quad x_{\text{orbit}} = x +
n, \quad y_{\text{orbit}} = y + n-1.
\end{equation}
The coadjoint orbits for discrete series have representatives
\begin{equation}
\lambda(x_{\text{orbit}},y_{\text{orbit}}) = \begin{cases} [ix_{\text{orbit}}e_1
-iy_{\text{orbit}}e_{p+1},i(x_{\text{orbit}} - y_{\text{orbit}})]&\\
\qquad x_{\text{orbit}} > 0 > y_{\text{orbit}}&\\[.5ex]
[ix_{\text{orbit}}e_1 +
(e_{2,p}-e_{p,2}+e_{2,p+1}+e_{p+1,2}),ix_{\text{orbit}}] &\\
\qquad x_{\text{orbit}} > 0 = y_{\text{orbit}}&\\[.5ex]
[ix_{\text{orbit}}e_1 + iy_{\text{orbit}}e_2,i(x_{\text{orbit}} -
y_{\text{orbit}})] & \\
\qquad x_{\text{orbit}} > y_{\text{orbit}} > 0 &
\end{cases}
\end{equation}
Then
\begin{equation}
\pi_{x,y}^{Sp(p,q)} = \pi(\text{orbit}\ \lambda(x_{\text{orbit}},y_{\text{orbit}})).
\end{equation}
(The partly nilpotent
coadjoint orbits $\lambda(x_{\text{orbit}},0)$ (with
$x_{\text{orbit}} >0$) can be treated as for $U(p,q)$.)
In the orbit method picture the condition \eqref{eq:Spbigx} simplifies to
\begin{equation}\label{eq:Spbigxorbit}
x_{\text{orbit}} > 0 > y_{\text{orbit}}.
\end{equation}
Similarly, \eqref{eq:Spsmallx} becomes
\begin{equation}\label{eq:Spsmallxorbit}
x_{\text{orbit}} > y_{\text{orbit}} \ge 0;
\end{equation}
equality in the inequality is the case of partially nilpotent coadjoint orbits.
In all cases we need also the genericity condition
\begin{equation}
\ell_{\text{orbit}} >0 \iff \ell > -(2n-1), \qquad
x_{\text{orbit}}-y_{\text{orbit}}>0 \iff x-y+1 > 0
\end{equation}
and the integrality conditions
\begin{equation}
x_{\text{orbit}} \equiv n \pmod{\mathbb Z},\qquad y_{\text{orbit}}
\equiv n-1 \pmod{\mathbb Z}.
\end{equation}
\end{subequations}
\begin{subequations}\label{se:Sppqirr}
Here is a sketch of proof of the irreducibility assertion from
Proposition \ref{prop:OSpbranch}. Each of the cohomologically induced
representations above is in the weakly fair range, so the general
theory of \cite{Vunit} applies. One conclusion of this theory is that
the cohomologically induced representations are irreducible modules
for a certain twisted differential operator algebra ${\mathcal
D}_{x,y}$; but in contrast to the $U(p,q)$ case, the natural map
$$U({\mathfrak s}{\mathfrak p}(p+q,{\mathbb C}) \times {\mathfrak
s}{\mathfrak p}(1,{\mathbb C})) \rightarrow {\mathcal D}_{x,y}$$
{\em need not} be surjective:
some of the cohomologically induced modules corresponding to discrete
series for $[Sp(2n,R)/Sp(2n-4,R)\times Sp(2,R)_\Delta$ are
{\em reducible}.
Here is an irreducibility proof for the case
\eqref{eq:Aq-Spsmallx}. We begin by defining
\begin{equation}
{\mathfrak q}_{0,\text{big}}^{Sp(p,q)\times Sp(1)} = {\mathfrak
l}_{0,\text{big}}^{Sp(p,q)\times Sp(1)} + {\mathfrak
u}_{0,\text{big}}^{Sp(p,q)\times Sp(1)} \supset
{\mathfrak q}_0^{Sp(p,q)\times Sp(1)}
\end{equation}
with Levi subgroup
\begin{equation}
L_{0,big}^{Sp(p,q)\times Sp(1)} = [U(2)_p\times Sp(p-2,q)] \times U(1).
\end{equation}
Define
\begin{equation}
\lambda^{0,\text{big}}_{x,y} = \left[\pi^{U(2)}_{x,y}\otimes
1\right]\otimes \xi_{x-y}.
\end{equation}
Induction by stages proves that
\begin{equation}
\pi^{Sp(p,q)\times Sp(1)}_{x,y} = A_{{\mathfrak q}^{Sp(p,q)\times
Sp(1)}_{0,\text{big}}}(\lambda^{0,\text{big}}_{x,y})
\qquad \ell/2 < x \le \ell + (n-1)).
\end{equation}
In this realization, the irreducibility argument from the $U(p,q)$
case goes through. The moment map from the cotangent bundle of the
(smaller) partial flag variety {\em is} birational onto its (normal)
image; so the map
\[U({\mathfrak s}{\mathfrak p}(p+q,{\mathbb C}) \times {\mathfrak
s}{\mathfrak p}(1,{\mathbb C})) \rightarrow {\mathcal
D}^{\text{small}}_{x,y}\]
is surjective, proving irreducibility. (The big parabolic subalgebra
defines a small partial flag variety, which is why we label the
twisted differential operator algebra ``small.'')
This argument does not apply to the case \eqref{eq:Aq-Spbigx},
since the corresponding larger Levi subgroup has a factor $U(1,1)$,
and the corresponding representation there is a discrete series. In
that case we have found only an unenlightening computational argument
for the irreducibility, which we omit.
Finally, the two series of derived functor modules fit together as
follows. If we consider the edge of the inequalities in
\eqref{eq:Spbigx} and \eqref{eq:Spsmallx}, namely
\[
(x,y) = \left ( \ell + (n-1), -(n-1) \right ),
\]
then we have
\begin{equation}\label{eq:Aq-Sp-coincide}
A_{{\mathfrak q}^{Sp(p,q)\times Sp(1)}_+}(\lambda^+_{x,y}) =
A_{{\mathfrak q}^{Sp(p,q)\times Sp(1)}_0}(\lambda^0_{x,y}).
\end{equation}
For this equality, as for the irreducibility of $A_{{\mathfrak
q}^{Sp(p,q)\times Sp(1)}_+}(\lambda^+_{x,y})$, we have found only
an unenlightening computational argument, which we omit.
\end{subequations}
| 3,499 | 71,259 |
en
|
train
|
0.35.19
|
\section{Octonionic hyperboloids}
\label{sec:spinp9-p}
\setcounter{equation}{0}
\begin{subequations}\label{se:spinp9-p}
We look for noncompact forms of the non-symmetric spherical space
$$S^{15} = \Spin(9)/\Spin(7)'$$
studied in Section \ref{sec:O}. The map from $\Spin(p,q)$ (with
$p+q=9$) to a form of
$O(16)$ will be given by the spin representation, which is therefore
required to be real. The spin representation is real if and only if
$p+q$ and $p-q$ are each congruent to $0$, $1$, or $7$ modulo $8$. The
candidates are
\begin{equation}
G=\Spin(5,4) \quad\text{or}\quad G=\Spin(8,1),
\end{equation}
with maximal compact subgroups
\begin{equation}
K = \Spin(5)\times_{\{\pm 1\}} \Spin(4) \quad\text{or}\quad \Spin(8);
\end{equation}
in the first case this means that the natural central subgroups $\{\pm 1\}$ in
$\Spin(5)$ and $\Spin(4)$ are identified with each other (and with the
natural central $\{\pm 1\}$ in $\Spin(5,4)$). In each case the
sixteen-dimensional spin representation of $G$ is real and preserves a
quadratic form of signature $(8,8)$. One way to see this is to notice
that the restriction of the spin representation to $K$ is a sum
of two irreducible representations
\begin{equation}
\spin(5) \otimes \spin(4)_{\pm} \quad \text{or} \quad \spin(8)_\pm
\end{equation}
Here $\spin(2m)_\pm$ denotes
the two half-spin representations, each of dimension $2^{m-1}$, of
$\Spin(2m)$. We are therefore looking at the hyperboloid
\begin{equation}\begin{aligned}
H_{8,8} &= \{v\in {\mathbb R}^{8,8} \mid \langle v,v\rangle_{8,8} = 1\}\\
&= \Spin(5,4)/\Spin(3,4)'\quad \text{or}\\
&=\Spin(8,1)/\Spin(7)'.
\end{aligned}\end{equation}
\end{subequations}
\begin{subequations}\label{se:spin81}
The discrete series for the second case was described
by Kobayashi in connection with branching from $SO(8,8)$ to $\Spin(8,1)$ in
\cite[Section 5.2]{toshi:howe}. Here we carry out an
approach
using the development above.
The harmonic analysis problem is
\begin{equation}
L^2(H_{8,8}) \simeq L^2(\Spin(8,1))^{\Spin(7)'};
\end{equation}
the $\Spin(7)'$ action is on the right. This problem is resolved by
Harish-Chandra's Plancherel formula for $\Spin(8,1)$: the discrete
series are exactly those of Harish-Chandra's discrete series that
contain a $\Spin(7)'$-fixed vector, and the multiplicity is the
dimension of that fixed space. Because of Helgason's branching law
from $\Spin(7)'$ to $\Spin(8)$ \eqref{eq:87'}, the number in question is
the sum of the multiplicities of the $\Spin(8)$ representations of
highest weights
\begin{equation}
\mu_y = (y/2,y/2,y/2,y/2) \qquad (y \in {\mathbb N}).
\end{equation}
Corollary \ref{cor:octdiff} constrains the possible infinitesimal
characters, and therefore the Harish-Chandra parameters, of
representations appearing on this hyperboloid. Here are the discrete
series having these infinitesimal characters. Suppose $x$ is an integer
satisfying $2x+y+7 >0$. Define
\begin{equation}
\pi^{\Spin(8,1)}_{x,y,\pm} = \begin{cases} \substack{\text{discrete
series with
parameter}\\((2x+y+7)/2,(y+5)/2,(y+3)/2,\pm(y+1)/2)}& x \ge 0\\[.5ex]
\qquad\qquad 0 & 0> x > -4\\[.5ex]
\substack{\text{discrete series with
parameter}\\ ((y+5)/2,(y+3)/2,(y+1)/2,\pm(2x+y+7)/2)} &
-4 \ge x > -(y+7)/2.
\end{cases} \end{equation}
We can now use Blattner's formula to determine which of these discrete
series contain $\Spin(8)$ representations of highest weight
$\mu_y$. The representations with a subscript $-$ are immediately
ruled out (since the last coordinate of the highest weight of any
$K$-type of such a discrete series must be negative). Similarly, in
the first case with $+$ the lowest $K$-type has highest weight
$(2x+1,1,1,1)+\mu_y$, and all other highest weights of $K$-types arise
by adding positive integers to these coordinates; so $\mu_y$ cannot
arise.
In the third case with $+$ the lowest $K$-type has highest weight
$(0,0,0,x+4)+\mu_y$; we get to $\mu_y$ by adding the nonnegative
multiple $-x-4$ of the noncompact positive root $e_4$. A more careful
examination of Blattner's formula shows that in fact $\mu_y$ has
multiplicity one. This proves
\begin{equation}\label{eq:spin81ds}
L^2(H_{8,8})_{\text{disc}} = \sum_{
y\ge 1,\ -4 \ge x > -(y+7)/2}
\pi^{\Spin(8,1)}_{x,y,+}.
\end{equation}
Furthermore (by Corollary \ref{cor:octdiff})
\begin{equation}
\pi^{O(8,8)}_\ell|_{\Spin(9,1)} = \sum_{\substack{
y\ge 1,\ -4 \ge x \ge -(y+7)/2\\[.4ex] 2x+y=\ell}}
\pi^{\Spin(8,1)}_{x,y,+}.
\end{equation}
These discrete series are cohomologically induced from
one-dimensional characters of the spin double cover of the compact
Levi subgroup
\begin{equation}
SO(2)\times U(3) \subset SO(2) \times SO(6) \subset SO(8) \subset
SO(8,1).
\end{equation}
\end{subequations}
\begin{subequations}\label{se:spin81orbit}
Here is the orbit method perspective. We have
\[\label{eq:spin81orbits}
({\mathfrak g}_0/{\mathfrak h}_0)^* \simeq \Spin^8 + {\mathbb R}^7
\]
as a representation of $H=\Spin(7)'$; the first summand is the
$8$-dimensional spin representation. What distinguishes this from the
compact case analyzed in \eqref{se:Oorbit} is that the restriction of
the natural $G$-invariant form has opposite signs on the two
summands; we take it to be negative on the first and positive on the
second. Because of \eqref{eq:GmodH},
the orbits we want are represented by $H$ orbits of maximal dimension
on this space. A generic orbit on ${\mathbb R}^7$ is given by the
value of the quadratic form
length $a_7 > 0$, and the corresponding isotropy group is
$\Spin(6)'\simeq SU(4)$. As a representation of $SU(4)$,
\[
\Spin^8 \simeq {\mathbb C}^4
\]
regarded as a real vector space. Here again the nonzero orbits are
indexed by the value of the Hermitian form $b_{\text{spin}} < 0$. The
conclusion is
that the regular $H$ orbits on $({\mathfrak g}_0/{\mathfrak h}_0)^*$
are
\[
\lambda(a_7,b_{\text{spin}}) \qquad (a_7> 0, \quad b_{\text{spin}} < 0).
\]
It turns out that the eigenvalues of such a matrix are $\pm i(a_7/4)^{1/2}$
(repeated three times), $\pm i(a_7/4 +b_{\text{spin}})^{1/2}$, and
one more eigenvalue zero. Accordingly the element is elliptic if and
only if $a_7/4 + b_{\text{spin}} \ge 0$. In this case we write
\[
x_{\text{orbit}} = (a_7/4 + b_{\text{spin}})^{1/2} -a_7^{1/2}/2, \quad
y_{\text{orbit}} = a^{1/2}_7 \qquad (a_7/4 + b_{\text{spin}} \ge 0)
\]
The elliptic elements we want are
\[
\lambda(x_{\text{orbit}},y_{\text{orbit}}) = (y_{\text{orbit}}/2,
y_{\text{orbit}}/2, y_{\text{orbit}}/2, y_{\text{orbit}}/2 +x), \qquad
(y_{\text{orbit}}/2 > -x_{\text{orbit}} > 0);
\]
we have represented the element (in fairly standard coordinates) by
something in the dual of a compact Cartan subalgebra $[{\mathfrak
s}{\mathfrak o}(2)]^4$ to which it is conjugate.
If now we define
\[
y = y_{\text{orbit}} -3, \quad x = x_{\text{orbit}} -2,
\]
then
\begin{equation}
\pi_{x,y,+}^{\Spin(8,1)} =
\pi(\text{orbit}\ \lambda(x_{\text{orbit}},y_{\text{orbit}})) \qquad
(0 > x_{\text{orbit}} > -y_{\text{orbit}}).
\end{equation}
When $y_{\text{orbit}} = 1$ or $2$ or $3$, or $x_{\text{orbit}}=-1$, these
representations are zero; that is the source of the conditions
$$y_{\text{orbit}} \ge 4, \quad -2 \ge x_{\text{orbit}} -
y_{\text{orbit}}/2$$
in \eqref{eq:spin81ds}.
\end{subequations}
\begin{subequations}\label{se:spin54}
In the first case of \eqref{se:spinp9-p}, we are looking at
\begin{equation}
H_{8,8} \simeq \Spin(5,4))/{\Spin(4,3)'};
\end{equation}
this is the ${\mathbb R}$-split version of Section \ref{sec:O}, and so
arises from
\begin{equation}
\Spin(4,3)' \ {\buildrel{\text{spin}}\over
\longrightarrow}\ \Spin(4,4) \subset \Spin(5,4).
\end{equation}
We have not determined the discrete series for this homogeneous space;
of course we expect two-parameter families of representations
cohomologically induced from one-dimensional characters of spin double
covers of real forms of $SO(2)\times U(3)$.
\end{subequations}
| 2,935 | 71,259 |
en
|
train
|
0.35.20
|
\section{The split $G_2$ calculation}
\label{sec:G2s}
\setcounter{equation}{0}
\begin{subequations}\label{se:G2hyp}
Write $G_{2,s}$ for the $14$-dimensional split Lie group
of type $G_2$. There is a $7$-dimensional real representation
$(\tau_{{\mathbb R},s},W_{{\mathbb R},s})$ of
$G_{2,s}$, whose weights are zero and the six short
roots. This preserves an inner product of signature $(4,3)$, and so
defines an inclusion
\begin{equation}\label{eq:G2SO7nc}
G_{2,s} \hookrightarrow SO(4,3).
\end{equation}
The corresponding actions of $G_{2,s}$ on the hyperboloids
\begin{equation}
H_{4,3} = O(4,3)/O(3,3), \qquad H_{3,4} = O(3,4)/O(2,4)
\end{equation}
are transitive. The isotropy groups are real forms of $SU(3)$:
\begin{equation}
H_{4,3} \simeq G_{2,s}/SL(3,{\mathbb R}), \qquad H_{3,4} \simeq
G_{2,s}/SU(2,1).
\end{equation}
\end{subequations}
The discrete series for these cases are given by Kobayashi (up to
two questions of reducibility) in
\cite[Thm 6.4]{Kob} ; see also \cite[Theorem 3.5]{toshi:zuckerman}.
We now give a self-contained treatment of the
classification, and resolve the reducibility.
\begin{subequations}\label{se:G2sreps}
The (real forms of) $O(7)$ representations appearing on these
hyperboloids are all related to the flag variety
\begin{equation}\begin{aligned}
O(7,{\mathbb C})/P &= \text{isotropic lines in\ }{\mathbb C}^7, \\
P = MN,\qquad M&=GL(1,{\mathbb C}) \times O(5,{\mathbb C}).
\end{aligned}\end{equation}
What makes everything simple is that $G_2({\mathbb C})$ is transitive
on this flag variety:
\begin{equation}\begin{aligned}
\text{isotropic lines in\ }{\mathbb C}^7 &= G_2({\mathbb C})/Q, \\
Q=LU, \qquad L &= GL(2,{\mathbb C}).
\end{aligned}\end{equation}
Precisely, the discrete series for $H_{4,3}$ are cohomologically
induced from the $\theta$-stable parabolic
\begin{equation}
{\mathfrak p}_1 = {\mathfrak m}_1 + {\mathfrak n}_1, \qquad
M_1=SO(2)\times O(2,3).
\end{equation}
The discrete series representations are
\begin{equation}
\pi^{O(4,3)}_{1,\ell} = A_{{\mathfrak p}_1}(\lambda_1(\ell)), \qquad \ell+5/2> 0.
\end{equation}
(cf. \eqref{se:Upq}). The inducing representation is the $SO(2)$
character indexed by $\ell$, and trivial on $O(2,3)$. Similarly, the
discrete series for $H_{3,4}$ are cohomologically
induced from the $\theta$-stable parabolic
\begin{equation}
{\mathfrak p}_2 = {\mathfrak m}_2 + {\mathfrak n}_2, \qquad
M_2=SO(2)\times O(1,4).
\end{equation}
The discrete series are
\begin{equation}
\pi^{O(3,4)}_{2,\ell} = A_{{\mathfrak p}_2}(\lambda_2(\ell)),\qquad \ell+5/2 > 0.
\end{equation}
The intersections of these parabolics with $G_2$ are
\begin{equation}
{\mathfrak q}_1 = {\mathfrak l}_1 + {\mathfrak u}_1, \qquad
L_1 = \text{long root $U(1,1)$}.
\end{equation}
and
\begin{equation}
{\mathfrak q}_2 = {\mathfrak l}_2 + {\mathfrak u}_2, \qquad
L_2= \text{long root $U(2)$}.
\end{equation}
(The Levi subgroups are just {\em locally} of this form.)
Because the $G_2$ actions on the $O(4,3)$ partial flag varieties are
transitive, we get discrete series representations for $H_{4,3}$
\begin{equation}
\pi^{G_{2,s}}_{1,\ell} = A_{{\mathfrak q}_1}(\lambda_1(\ell)),\qquad \ell+5/2 > 0.
\end{equation}
The character is $\ell$ times the action of $L_1$ on the highest short
root defining ${\mathfrak q}_1$.
Similarly, for the action on $H_{3,4}$
\begin{equation}
\pi^{G_{2,s}}_{2,\ell} = A_{{\mathfrak q}_2}(\lambda_2(\ell)),\qquad \ell +5/2 > 0.
\end{equation}
The {\tt atlas} software \cite{atlas} tells us that all of these
discrete series representations of $G_2$ are irreducible, with the
single exception of $\pi^{G_{2,s}}_{1,-2} = A_{{\mathfrak
q}_1}(\lambda_1(-2)).$ That representation is a sum of two
irreducible constituents. One constituent is the unique non-generic
limit of discrete series of infinitesimal character a short root. In
\cite{G2}*{Theorem 18.5}, (describing some of Arthur's unipotent
representations) this is the representation described in (b). The
other constituent is described in part (c) of that same theorem. The
irreducible representation $\pi^{G_{2,s}}_{2,-2} = A_{{\mathfrak
q}_2}(\lambda_2(-2))$ appears in part (a) of the
theorem. All of these identifications (including the reducibility of
$\pi^{G_{2,s}}_{1,-2}$) follow from knowledge of the
$K$-types of these representations (given in \eqref{se:G2sK} below)
and the last assertion of \cite{G2}*{Theorem 18.5}.
Summarizing, in the notation of \cite{G2},
\begin{equation}\label{eq:G2unip}
\pi^{G_{2,s}}_{1,-2} \simeq J_{-}(H_2; (2, 0)) \oplus J(H_2; (1,1)),
\qquad \pi^{G_{2,s}}_{2,-2} \simeq J(H_1; (1,1)).
\end{equation}
That is, the first discrete series for these non-symmetric spherical
spaces include three of the five unipotent representations for the
split $G_2$ attached to the principal nilpotent in $SL(3) \subset
G_2$.
\end{subequations}
\begin{subequations}\label{G2sorbit}
Here is the orbit method perspective. For the case of $H_{4,3}$, the
representation of $H=SL(3,{\mathbb R})$ on $[{\mathfrak
g}_0/{\mathfrak h}_0]^*$ is ${\mathbb R}^3 + ({\mathbb
R}^3)^*$. The generic orbits of $H$ are indexed by non-zero real
numbers $A$, the value of a linear functional on a vector. We can
arrange the normalizations so that the elliptic elements are exactly
those with $A>0$; if we define
\[ \ell_{\text{orbit}} = A^{1/2}, \qquad \ell = \ell_{\text{orbit}} - 5/2, \]
and write $\lambda_1(\ell_{\text{orbit}})$ for a representative of this
orbit, then
\[
\pi_{1,\ell}^{G_{2,s}} =
\pi({\text{orbit}},\lambda_1(\ell_{\text{orbit}})) \qquad
(\ell_{\text{orbit}} > 0).
\]
For the case of $H_{3,4}$, the representation of $H=SU(2,1)$ on $[{\mathfrak
g}_0/{\mathfrak h}_0]^*$ is ${\mathbb C}^{2,1}$; generic orbits
are parametrized by the nonzero values $B$ of the Hermitian form of signature
$(2,1)$. The elliptic orbits are those with $B>0$; if we define
\[ \ell_{\text{orbit}} = B^{1/2}, \qquad \ell = \ell_{\text{orbit}} - 5/2 \]
then
\[
\pi_{2,\ell}^{G_{2,s}} =
\pi({\text{orbit}},\lambda_2(\ell_{\text{orbit}})) \qquad
(\ell_{\text{orbit}} > 0).
\]
\end{subequations}
\begin{subequations} \label{se:G2sK}
We conclude this section by calculating the restrictions to
\begin{equation}
K = SU(2)_{\text{long}} \times_{\{\pm 1\}} SU(2)_{\text{short}}
\subset G_{2,s}.
\end{equation}
We define
\begin{equation}\begin{aligned}
\gamma_d^{\text{long}} &= \text{$(d+1)$-diml irr of
$SU(2)_{\text{long}}$}\\
\gamma_d^{\text{short}} &= \text{$(d+1)$-diml irr of
$SU(2)_{\text{short}}$}
\end{aligned}\end{equation}
The maximal compact of $O(4,3)$ is $O(4)\times O(3)$. The embedding of
$G_{2,s}$ sends $SU(2)_{\text{long}}$ to one of the factors in
$$O(4) \supset SO(4) \simeq SU(2)\times_{\{\pm 1\}} SU(2),$$
and sends $SU(2)_{\text{short}}$ diagonally into the product of the
other $SU(2)$ factor and $SO(3)\subset O(3)$ (by the two-fold cover
$SU(2) \rightarrow SO(3)$). According to \eqref{eq:Obranch},
\begin{equation}\begin{aligned}
\pi^{O(4,3)}_{1,\ell}|_{O(4)\times O(3)} = \sum_{\substack{d-\ell-3
\ge e \ge 0 \\[.2ex] e\equiv d-\ell-3 \pmod{2}}} \pi^{O(4)}_d \otimes
\pi^{O(3)}_e\\
\pi^{O(3,4)}_{2,\ell}|_{O(3)\times O(4)} = \sum_{\substack{d' -\ell -4
\ge e' \ge 0\\[.2ex] e'\equiv d'-\ell-4\pmod{2}}} \pi^{O(3)}_{d'} \otimes
\pi^{O(3)}_{e'}.
\end{aligned}\end{equation}
By an easy calculation, we deduce
\begin{equation}\begin{aligned}
\pi^{G_{2,s}}_{1,\ell}|_K = \sum_{\substack{d-\ell-3 \ge e\ge 0\\[.2ex]
e\equiv d-\ell-3 \pmod{2}}} \gamma_d^{\text{long}} \otimes
\left[\gamma_d^{\text{short}} \otimes
\gamma_{2e}^{\text{short}}\right].\\
\pi^{G_{2,s}}_{2,\ell}|_K = \sum_{\substack{d'-\ell-4 \ge e'\ge
0\\[.2ex] e'\equiv d'-\ell-4\pmod{2}}}
\gamma^{\text{long}}_{e'} \otimes \left[ \gamma_{e'}^{\text{short}} \otimes
\gamma_{2d'}^{\text{short}}\right]
\end{aligned}\end{equation}
The internal tensor products in the short $SU(2)$ factors are of
course easy to compute:
\begin{equation}
\pi^{G_{2,s}}_{1,\ell}|_K = \sum_{\substack{d-\ell-3 \ge e\ge
0\\[.2ex] e\equiv d-\ell-3\pmod{2}}} \sum_{k=0}^{\min(d,2e)}
\gamma^{\text{long}}_{e'} \otimes \gamma_{d+2e-2k}^{\text{short}},
\end{equation}
\begin{equation}
\pi^{G_{2,s}}_{2,\ell}|_K = \sum_{\substack{d'-\ell-4 \ge e'\ge
0\\[.2ex] e'\equiv d'-\ell-4\pmod{2}}} \sum_{k'=0}^{e'}
\gamma^{\text{long}}_{e'} \otimes \gamma_{2d'+e'-2k'}^{\text{short}}
\end{equation}
\end{subequations}
| 3,248 | 71,259 |
en
|
train
|
0.35.21
|
\section{The noncompact big $G_2$ calculation}
\label{sec:bigncG2}
\setcounter{equation}{0}
\begin{subequations}\label{se:bigncG2hyp}
In this section we look at noncompact forms of $S^7 \simeq
\Spin(7)'/G_{2,c}$ from Section \ref{sec:bigG2}. The noncompact forms
of $\Spin(7)$ are $\Spin(p,q)$ with $p+q=7$, having maximal compact
subgroups $\Spin(p)\times_{\{\pm 1\}} \Spin(q)$. None of these compact
subgroups can contain $G_{2,c}$ (unless $pq=0$), so the isotropy subgroup we are
looking for is the split form $G_{2,s}$. The seven-dimensional
representation of $G_{2,s}$ is real, and its invariant bilinear form
is of signature $(3,4)$; so we are looking at
\begin{equation}
G_{2,s} \hookrightarrow \Spin(3,4),
\end{equation}
the double cover of the inclusion \eqref{eq:G2SO7nc}. This homogeneous
space is discussed briefly in \cite{Kob}*{Corollary 5.6(e)}, which is proven
in part (ii) of the proof on page 197. We will argue along similar
lines, but get more complete conclusions (parallel to
Kobayashi's results described in Sections
\ref{sec:Upq}--\ref{sec:Sppq}).
The
eight-dimensional spin representation of $\Spin(3,4)$ is real and of
signature $(4,4)$, so we get
\begin{equation}
\Spin(3,4)' \hookrightarrow \Spin(4,4), \qquad \Spin(3,4)'\cap
\Spin(3,4) = G_{2,s}.
\end{equation}
The $\Spin(3,4)'$ action on
\begin{equation}
H_{4,4} = \Spin(4,4)/\Spin(3,4)
\end{equation}
is transitive, so
\begin{equation}
H_{4,4}\simeq \Spin(3,4)'/G_{2,s}.
\end{equation}
In a similar fashion, we find an identification of six-dimensional
complex manifolds
\begin{equation}\label{eq:qmatchbigG2}
\Spin(4,4)/[\Spin(2)\times_{\{\pm 1\}}] \Spin(2,4)] \simeq
\Spin(3,4)'/\widetilde{U(1,2)}.
\end{equation}
The manifold on the left corresponds to the $\theta$-stable parabolic
${\mathfrak q}^{O(4,4)}$ described in \eqref{eq:qOpq}; the discrete
series $\pi^{O(4,4)}_\ell$ for $H_{4,4}$ are obtained from it by
cohomological induction.
The manifold on the right corresponds to the $\theta$-stable parabolic
\begin{equation}\label{eq:qSpin34}
{\mathfrak q}^{\Spin(3,4)'} = {\mathfrak l}^{\Spin(3,4)'} +
{\mathfrak u}^{\Spin(3,4)'} \subset
{\mathfrak o}(7,{\mathbb C});
\end{equation}
the corresponding Levi subgroup is
\begin{equation}
L^{\Spin(3,4)'} = \widetilde{U(1,2)}
\end{equation}
The covering here is the ``square root of determinant'' cover; the
one-dimensional characters are half integer powers of the
determinant. We are interested in
\begin{equation}\begin{aligned}
\lambda_\ell &= {\det}^{\ell/2} \in [L^{\Spin(3,4)'}]\,\widehat{\ }
\qquad (\ell +3 > 0).\\
\pi^{\Spin(3,4)'}_\ell &= A_{{\mathfrak q}^{\Spin(3,4)}}(\lambda_\ell) \qquad
(\ell > -3).
\end{aligned}\end{equation}
The infinitesimal character of this representation is
\begin{equation}
\text{infl char}(\pi_\ell^{\Spin(4,3)'}) = ((\ell+5)/2,(\ell+3)/2,(\ell+1)/2).
\end{equation}
As a consequence of \eqref{eq:qmatchbigG2},
\begin{equation}
\pi^{O(4,4)}_\ell|_{\Spin(3,4)'} \simeq \pi^{\Spin(3,4)'}_\ell.
\end{equation}
The discrete part of the Plancherel decomposition is therefore
\begin{equation}\label{eq:bigncG2disc}
L^2(H_{4,4})_{\text{disc}} = \sum_{\ell > -3}
\pi_\ell^{\Spin(3,4)'}.
\end{equation}
The ``weakly fair'' range for $\pi^{\Spin(3,4)'}_\ell$ is $\ell \ge
-3$, so all the representations $\pi^{\Spin(3,4)'}_\ell$ are
contained in the weakly fair range. In particular, \cite{Vunit}
establishes {\em a priori} the unitarity of what turn out to be the
discrete series representations.
But the results in \cite{Vunit} prove only
\begin{equation}\label{eq:bigG2unit}
\text{$\pi^{\Spin(3,4)'}_\ell$ is irreducible for $\ell \ge 0$.}
\end{equation}
The {\tt atlas} software \cite{atlas} proves the irreducibility of the
first two discrete series (those not covered by \eqref{eq:bigG2unit}).
\end{subequations}
\begin{subequations}\label{se:bigG2sorbit}
Here is the orbit method perspective. The
representation of $H=G_{2,s}$ on $[{\mathfrak
g}_0/{\mathfrak h}_0]^*$ is ${\mathbb R}^{3,4}$, the real
representation whose highest weight is a short
root. We have already said that this representation carries an
invariant quadratic form of signature $(3,4)$. The generic orbits of
$H$ are indexed by non-zero real
numbers $A$, the values of the quadratic form. We can
arrange the normalizations so that the elliptic elements are exactly
those with $A>0$; if we define
\[ \ell_{\text{orbit}} = A^{1/2}, \qquad \ell = \ell_{\text{orbit}} - 3, \]
and write $\lambda(\ell_{\text{orbit}})$ for a representative of this
orbit, then
\[
\pi_\ell^{\Spin(3,4)} =
\pi({\text{orbit}},\lambda(\ell_{\text{orbit}})) \qquad
(\ell_{\text{orbit}} > 0).
\]
\end{subequations}
\begin{bibdiv}
\begin{biblist}[\normalsize]
\bib{atlas}{misc}{
title={Atlas of Lie Groups and Representations {\rm software package}},
year={2017},
note={URL: {\tt http://www.liegroups.org}},
}
\bib{BVUpq}{incollection}
{
AUTHOR = {Barbasch, Dan},
AUTHOR = {Vogan, David},
TITLE = {Weyl group representations and nilpotent orbits},
BOOKTITLE = {Representation theory of reductive groups ({P}ark {C}ity,
{U}tah, 1982)},
SERIES = {Progr. Math.},
VOLUME = {40},
PAGES = {21--33},
PUBLISHER = {Birkh\"auser Boston, Boston, MA},
YEAR = {1983},
}
\bib{BOct}{article}{
author={Borel, Armand},
title={Le plan projectif des octaves et les sph\`eres comme espaces
homog\`enes},
language={French},
journal={C. R. Acad. Sci. Paris},
volume={230},
date={1950},
pages={1378--1380},
}
\bib{BHomog}{article}{
author={Borel, Armand},
title={Sur la cohomologie des espaces fibr\'es principaux et des espaces
homog\`enes de groupes de Lie compacts},
language={French},
journal={Ann. of Math. (2)},
volume={57},
date={1953},
pages={115--207},
}
\bib{Hinvt}{article}{
author={Helgason, Sigurdur},
title={Differential operators on homogeneous spaces},
journal={Acta Math.},
volume={102},
date={1959},
pages={239--299},
}
\bib{HinvtBull}{article}{
author={Helgason, Sigurdur},
title={Invariant differential equations on homogeneous manifolds},
journal={Bull. Amer. Math. Soc.},
volume={83},
date={1977},
number={5},
pages={751--774},
}
\bib{GGA}{book}{
author={Helgason, Sigurdur},
title={Groups and geometric analysis},
series={Mathematical Surveys and Monographs},
volume={83},
publisher={American Mathematical Society, Providence, RI},
date={2000},
pages={xxii+667},
}
\bib{KBeyond}{book}{
author={Knapp, Anthony W.},
title={Lie Groups Beyond an Introduction},
edition={second edition},
series={Progress in Mathematics 140},
publisher={Birkh\"auser},
address={Boston-Basel-Berlin},
date={2002},
}
\bib{Kobayashi-Stiefel}{article}{
AUTHOR = {Kobayashi, Toshiyuki},
TITLE = {Singular unitary representations and discrete series for
indefinite {S}tiefel manifolds {${\rm U}(p,q;{\bf F})/{\rm
U}(p-m,q;{\bf F})$}},
JOURNAL = {Mem. Amer. Math. Soc.},
VOLUME = {95},
YEAR = {1992},
NUMBER = {462},
PAGES = {vi+106},
}
\bib{Kob}{article}{
author={Kobayashi, Toshiyuki},
title={Discrete decomposability of the restriction of $A_{\germ
q}(\lambda)$ with respect to reductive subgroups and its applications},
journal={Invent. Math.},
volume={117},
date={1994},
number={2},
pages={181--205},
}
\bib{toshi:zuckerman}{article}
{ AUTHOR = {Kobayashi, Toshiyuki},
TITLE = {Branching problems of {Z}uckerman derived functor modules},
conference={
title={Representation Theory and Mathematical Physics},
},
book ={
title={Contemp. Math.},
VOLUME = {557},
PUBLISHER = {Amer. Math. Soc., Providence, RI},
},
date = {2011},
PAGES = {23--40},
}
\bib{toshi:howe}{article}{
AUTHOR = {Kobayashi, Toshiyuki},
TITLE = {Global analysis by hidden symmetry},
conference={title={Representation Theory, Number Theory, and Invariant Theory}},
book = {
title= {Progr. Math.},
VOLUME = {323},
}
PAGES = {359--397},
PUBLISHER = {Birkh\"auser/Springer},
date= {2017},
}
\bib{invt}{article}{
author={Koornwinder, Tom H.},
title={Invariant differential operators on nonreductive homogeneous
spaces},
pages={i+15},
eprint={arXiv:math/0008116 [math.RT]},
}
\bib{KKOS}{article}{
author={Kr\"otz, Bernhard},
author={Kuit, Job J.},
author={Opdam, Eric M.},
author={Schlichtkrull, Henrik},
title={The infinitesimal characters of discrete series for real spherical spaces},
date={2017},
pages={40},
eprint={arXiv:1711.08635 [math RT]},
}
\bib{MS}{article}{
author={Montgomery, Deane},
author={Samelson, Hans},
title={Transformation groups of spheres},
journal={Ann. of Math. (2)},
volume={44},
date={1943},
pages={454--470},
}
\bib{Oni69}{article}{
author={Oni\v s\v cik, A. L.},
title={Decompositions of reductive Lie groups},
language={Russian},
journal={Mat. Sb. (N.S.)},
volume={80 (122)},
date={1969},
pages={553--599},
translation={
journal={Math. USSR Sb.},
volume={9},
date={1969},
pages={515--554},
},
}
\bib{OM}{article}{
author={\=Oshima, Toshio},
author={Matsuki, Toshihiko},
title={A description of discrete series for semisimple symmetric spaces},
conference={
title={Group representations and systems of differential equations},
address={Tokyo},
date={1982},
},
book={
series={Adv. Stud. Pure Math.},
volume={4},
publisher={North-Holland, Amsterdam},
},
date={1984},
pages={331--390},
}
\bib{RossH}{article}{
author={Rossmann, Wulf},
title={Analysis on real hyperbolic spaces},
journal={J. Funct. Anal.},
volume={30},
date={1978},
number={3},
pages={448--477},
}
\bib{SR}{article}{
author={S.~Salamanca-Riba},
title={On the unitary dual of real semisimple Lie groups snd the
$A_{\mathfrak q}(\lambda)$ modules: the strongly regular case},
journal={Duke Math.\ J.},
volume={96},
date={1999},
pages={521--546},
}
\bib{StrH}{article}{
author={Strichartz, Robert S.},
title={Harmonic analysis on hyperboloids},
journal={J. Functional Analysis},
volume={12},
date={1973},
pages={341--383},
}
\bib{VGLn}{article}{
author={Vogan, David A., Jr.},
TITLE = {The unitary dual of {${\rm GL}(n)$} over an {A}rchimedean
field},
JOURNAL = {Invent. Math.},
VOLUME = {83},
YEAR = {1986},
NUMBER = {3},
PAGES = {449--505},
}
| 4,038 | 71,259 |
en
|
train
|
0.35.22
|
\bib{Kobayashi-Stiefel}{article}{
AUTHOR = {Kobayashi, Toshiyuki},
TITLE = {Singular unitary representations and discrete series for
indefinite {S}tiefel manifolds {${\rm U}(p,q;{\bf F})/{\rm
U}(p-m,q;{\bf F})$}},
JOURNAL = {Mem. Amer. Math. Soc.},
VOLUME = {95},
YEAR = {1992},
NUMBER = {462},
PAGES = {vi+106},
}
\bib{Kob}{article}{
author={Kobayashi, Toshiyuki},
title={Discrete decomposability of the restriction of $A_{\germ
q}(\lambda)$ with respect to reductive subgroups and its applications},
journal={Invent. Math.},
volume={117},
date={1994},
number={2},
pages={181--205},
}
\bib{toshi:zuckerman}{article}
{ AUTHOR = {Kobayashi, Toshiyuki},
TITLE = {Branching problems of {Z}uckerman derived functor modules},
conference={
title={Representation Theory and Mathematical Physics},
},
book ={
title={Contemp. Math.},
VOLUME = {557},
PUBLISHER = {Amer. Math. Soc., Providence, RI},
},
date = {2011},
PAGES = {23--40},
}
\bib{toshi:howe}{article}{
AUTHOR = {Kobayashi, Toshiyuki},
TITLE = {Global analysis by hidden symmetry},
conference={title={Representation Theory, Number Theory, and Invariant Theory}},
book = {
title= {Progr. Math.},
VOLUME = {323},
}
PAGES = {359--397},
PUBLISHER = {Birkh\"auser/Springer},
date= {2017},
}
\bib{invt}{article}{
author={Koornwinder, Tom H.},
title={Invariant differential operators on nonreductive homogeneous
spaces},
pages={i+15},
eprint={arXiv:math/0008116 [math.RT]},
}
\bib{KKOS}{article}{
author={Kr\"otz, Bernhard},
author={Kuit, Job J.},
author={Opdam, Eric M.},
author={Schlichtkrull, Henrik},
title={The infinitesimal characters of discrete series for real spherical spaces},
date={2017},
pages={40},
eprint={arXiv:1711.08635 [math RT]},
}
\bib{MS}{article}{
author={Montgomery, Deane},
author={Samelson, Hans},
title={Transformation groups of spheres},
journal={Ann. of Math. (2)},
volume={44},
date={1943},
pages={454--470},
}
\bib{Oni69}{article}{
author={Oni\v s\v cik, A. L.},
title={Decompositions of reductive Lie groups},
language={Russian},
journal={Mat. Sb. (N.S.)},
volume={80 (122)},
date={1969},
pages={553--599},
translation={
journal={Math. USSR Sb.},
volume={9},
date={1969},
pages={515--554},
},
}
\bib{OM}{article}{
author={\=Oshima, Toshio},
author={Matsuki, Toshihiko},
title={A description of discrete series for semisimple symmetric spaces},
conference={
title={Group representations and systems of differential equations},
address={Tokyo},
date={1982},
},
book={
series={Adv. Stud. Pure Math.},
volume={4},
publisher={North-Holland, Amsterdam},
},
date={1984},
pages={331--390},
}
\bib{RossH}{article}{
author={Rossmann, Wulf},
title={Analysis on real hyperbolic spaces},
journal={J. Funct. Anal.},
volume={30},
date={1978},
number={3},
pages={448--477},
}
\bib{SR}{article}{
author={S.~Salamanca-Riba},
title={On the unitary dual of real semisimple Lie groups snd the
$A_{\mathfrak q}(\lambda)$ modules: the strongly regular case},
journal={Duke Math.\ J.},
volume={96},
date={1999},
pages={521--546},
}
\bib{StrH}{article}{
author={Strichartz, Robert S.},
title={Harmonic analysis on hyperboloids},
journal={J. Functional Analysis},
volume={12},
date={1973},
pages={341--383},
}
\bib{VGLn}{article}{
author={Vogan, David A., Jr.},
TITLE = {The unitary dual of {${\rm GL}(n)$} over an {A}rchimedean
field},
JOURNAL = {Invent. Math.},
VOLUME = {83},
YEAR = {1986},
NUMBER = {3},
PAGES = {449--505},
}
\bib{Vunit}{article}{
author={Vogan, David A., Jr.},
title={Unitarizability of certain series of representations},
journal={Ann.\ of Math.},
volume={120},
date={1984},
pages={141--187},
}
\bib{Virr}{article}{
author={Vogan, David A., Jr.},
title={Irreducibility of discrete series representations for semisimple
symmetric spaces},
conference={
title={Representations of Lie groups, Kyoto, Hiroshima, 1986},
},
book={
series={Adv. Stud. Pure Math.},
volume={14},
publisher={Academic Press, Boston, MA},
},
date={1988},
pages={191--221},
}
\bib{G2}{article}{
author={Vogan, David A., Jr.},
title={The unitary dual of $G_2$},
journal={Invent.\ Math.},
volume={116},
date={1994},
pages={677--791},
}
\bib{VZ}{article}{
author={Vogan, David A., Jr.},
author={Zuckerman, Gregg},
title={Unitary representations with non-zero cohomology},
journal={Compositio Math.},
volume={53},
date={1984},
pages={51--90},
}
\bib{Wolfflag}{article}{
author={J. Wolf},
title={The action of a real semisimple group on a complex flag
manifold. I. Orbit structure and holomorphic arc components},
journal={Bull.\ Amer.\ Math.\ Soc.},
volume={75},
date={1969},
pages={1121--1237},
}
\bib{Wolf}{book}{
author={Wolf, Joseph A.},
title={Harmonic analysis on commutative spaces},
series={Mathematical Surveys and Monographs},
volume={142},
publisher={American Mathematical Society, Providence, RI},
date={2007},
pages={xvi+387},
}
\end{biblist}
\end{bibdiv}
\end{document}
| 2,078 | 71,259 |
en
|
train
|
0.36.0
|
\begin{document}
\title{On crossed products of the Cuntz algebra ${\mathcal O}_\infty$ by quasi-free actions of abelian groups}
\author{Takeshi KATSURA\\
Department of Mathematical Sciences\\
University of Tokyo, Komaba, Tokyo, 153-8914, JAPAN\\
e-mail: {\tt [email protected]}}
\date{}
\maketitle
\begin{abstract}
{\footnotesize We investigate the structures of crossed products of the
Cuntz algebra ${\mathcal O}_\infty$ by quasi-free actions of abelian groups.
We completely determine their ideal structures and compute the strong Connes
spectra and K-groups.}
\varepsilonnd{abstract}
\section{Introduction}
The crossed products of $C^*$-algebras give us plenty of interesting examples,
and the structures of them have been examined by several authors.
In \cite{Ki}, A. Kishimoto gave a necessary and sufficient condition
that the crossed products by abelian groups become simple
in terms of the strong Connes spectrum.
For the case of the crossed products of Cuntz algebras
by so-called quasi-free actions of abelian groups,
he gave a condition for simplicity, which is easy to check.
In \cite{KK1} and \cite{KK2}, A. Kishimoto and A. Kumjian dealt with,
among others, the crossed products of Cuntz algebras by quasi-free actions of
the real group $\mathbb{R}$.
In our previous papers \cite{Ka1}, \cite{Ka2},
we examined the structures of crossed products of Cuntz algebras ${\mathcal O}_n$
by quasi-free actions of
arbitrary locally compact, second countable, abelian groups.
The class of our algebras has many examples of simple stably projectionless
$C^*$-algebras as well as AF-algebras and purely infinite $C^*$-algebras.
In \cite{Ka1}, we completely determined the ideal structures of our algebras,
and gave another proof of A. Kishimoto's result on the simplicity of them.
We also gave a necessary and sufficient condition that our algebras become
primitive, and computed the Connes spectra and K-groups of our algebras.
In \cite{Ka2}, we proved that our algebras become AF-embeddable
when actions satisfy certain conditions.
To the best of the author's knowledge, this is the first case
to have succeeded in embedding crossed products of purely infinite
$C^*$-algebras into AF-algebras except trivial cases.
We also gave a necessary and sufficient condition that our algebras become
simple and purely infinite,
and consequently our algebras are either purely infinite or AF-embeddable
when they are simple.
In this paper, we deal with crossed products of the Cuntz algebra ${\mathcal O}_\infty$
by quasi-free actions of arbitrary locally compact, second countable,
abelian groups.
From section 3 to section 6, we completely determine the ideal structures
of such algebras by using the technique developed in \cite{Ka1}.
We omit detailed computations if similar computations have been already done
in \cite{Ka1}.
Readers are referred to \cite{Ka1}.
In the last section, we gather some results on crossed products
of the Cuntz algebra ${\mathcal O}_\infty$.
Among others, we give another proof of the determination of the simplicity
of the crossed products done by A. Kishimoto,
and we succeed in computing the strong Connes spectra of quasi-free actions
on the Cuntz algebra ${\mathcal O}_\infty$.
The crossed products examined in this paper or in \cite{Ka1}, \cite{Ka2},
can be considered as continuous counterparts of
Cuntz-Krieger algebras or graph algebras (cf. \cite{D}).
From this point of view, the crossed products of ${\mathcal O}_n$ can be considered
as graph algebras of locally finite graphs,
and the ones of ${\mathcal O}_\infty$ can be considered as graph algebras of graphs
whose vertices emit and receive infinitely many edges.
Recently the ideal structures of graph algebras, which is not necessarily
locally finite, were deeply examined in \cite{BHRS} and \cite{HS}.
Compared with row finite case,
it is rather difficult to describe ideal structures of graph algebras
which have vertices emitting infinitely many edges.
This seems to be related to the difficulty of examination of the
ideal structures of the crossed products of ${\mathcal O}_\infty$
compared with the ones of ${\mathcal O}_n$ done in \cite{Ka1}.
{\bf Acknowledgment.}
The author would like to thank
to his advisor Yasuyuki Kawahigashi for his support and encouragement,
to Masaki Izumi for various comments and many suggestions.
He is also grateful to Iain Raeburn and Wojciech Szyma\'nski
for stimulating discussions.
This work was partially supported by Research Fellowship
for Young Scientists of the Japan Society for the Promotion of Science.
\section{Preliminaries}\label{PRE}
The Cuntz algebra ${\mathcal O}_\infty$ is the universal $C^*$-algebra generated
by infinitely many isometries
$S_1,S_2,\ldots$ satisfying $S_i^*S_j=\delta_{i,j}$.
For $n\in\mathbb{Z}_+:=\{1,2,\ldots\}$ and $k\in\mathbb{N}:=\{0,1,\ldots\}$,
we define the set ${\mathcal W}_n^{(k)}$ of words in $\{1,2,\ldots,n\}$
with length $k$ by ${\mathcal W}_n^{(0)}=\{\varepsilonmptyset\}$ and
$${\mathcal W}_n^{(k)}
=\big\{ (i_1,i_2,\ldots,i_k)\ \big|\ i_j\in\{1,2,\ldots,n\}\big\}$$
for $k\geq 1$.
Set ${\mathcal W}_n=\bigcup_{k=0}^\infty {\mathcal W}_n^{(k)}$ and
${\mathcal W}_\infty=\bigcup_{n=1}^\infty {\mathcal W}_n$.
For $\mu=(i_1,i_2,\ldots,i_k)\in{\mathcal W}_\infty$,
we denote its length $k$ by $|\mu|$,
and set $S_\mu=S_{i_1}S_{i_2}\cdots S_{i_k}\in{\mathcal O}_\infty$.
Let $G$ be a locally compact abelian group which satisfies
the second axiom of countability and ${\mathcal G}amma$ be the dual group of $G$.
We use $+$ for multiplicative operations of abelian groups except for $\mathbb{T}$,
which is the group of the unit circle in the complex plane $\mathbb{C}$.
The pairing of $t\in G$ and $\gamma\in{\mathcal G}amma$ is denoted by
$\ip{t}{\gamma}\in\mathbb{T}$.
For $\omega=(\omega_1,\omega_2,\ldots)\in{\mathcal G}amma^\infty$,
we define an action $\alpha^\omega$ of abelian group $G$ on ${\mathcal O}_\infty$ by
$\alpha^\omega_t(S_i)=\ip{t}{\omega_i}S_i$ for $i\in\mathbb{Z}_+$ and $t\in G$.
The action $\alpha^\omega:G\curvearrowright{\mathcal O}_\infty$ becomes quasi-free
(for a definition of quasi-free actions on Cuntz algebras, see \cite{E}).
However, there exist quasi-free actions of abelian group $G$ on ${\mathcal O}_\infty$,
which are not conjugate to $\alpha^\omega$ for any $\omega\in{\mathcal G}amma^\infty$
though we do not deal with such actions.
The crossed product ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ has a $C^*$-subalgebra
$\mathbb{C} 1{\rtimes_{\alpha^\omega}}G$ which is isomorphic to $C_0({\mathcal G}amma)$.
We consider $C_0({\mathcal G}amma)$ as a $C^*$-subalgebra of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$.
The Cuntz algebra ${\mathcal O}_\infty$ is naturally embedded into
the multiplier algebra $M({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)$ of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$.
For each $\mu=(i_1,i_2,\ldots,i_k)\in{\mathcal W}_\infty$,
we define an element $\omega_\mu$ of ${\mathcal G}amma$
by $\omega_\mu=\sum_{j=1}^{k}\omega_{i_j}$.
For $\gamma_0\in{\mathcal G}amma$, we define a (reverse) shift automorphism
$\sigma_{\gamma_0}:C_0({\mathcal G}amma)\to C_0({\mathcal G}amma)$ by
$(\sigma_{\gamma_0} f)(\gamma)=f(\gamma+\gamma_0)$ for $f\in C_0({\mathcal G}amma)$.
Once noting that $\alpha^\omega_t(S_\mu)=\ip{t}{\omega_\mu}S_\mu$
for $\mu\in{\mathcal W}_\infty$, one can easily verify that
$fS_\mu =S_\mu\sigma_{\omega_\mu}f$ for any $f\in C_0({\mathcal G}amma)\subset {\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$.
For a subset $X$ of a $C^*$-algebra,
we denote by $\spa X$ the linear span of $X$,
and by $\cspa X$ its closure.
We have
${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G=\cspa\{ S_\mu fS_\nu^*\mid \mu,\nu\in{\mathcal W}_\infty,\ f\in C_0({\mathcal G}amma)\}$.
We denote by $\mathbb{M}_k$ the $C^*$-algebra of $k \times k$ matrices
for $k=1,2,\ldots$,
and by $\mathbb{K}$ the $C^*$-algebra of compact operators
of the infinite dimensional separable Hilbert space.
| 2,530 | 50,987 |
en
|
train
|
0.36.1
|
\section{Gauge invariant ideals}
In this section, we determine all the ideals
which are globally invariant under the gauge action.
Here an ideal means a closed two-sided ideal,
and the gauge action $\beta:\mathbb{T}\curvearrowright{\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ is defined by
$\beta_t(S_\mu fS_\nu^*)=t^{|\mu|-|\nu|}S_\mu fS_\nu^*$
for $\mu,\nu\in{\mathcal W}_\infty,\ f\in C_0({\mathcal G}amma)$ and $t\in\mathbb{T}$.
For a positive integer $n$, we define a projection $p_n$ by
$p_n=1-\sum_{i=1}^nS_iS_i^*$.
We set $p_0=1$.
Since $p_n$ commutes with $C_0({\mathcal G}amma)$,
$p_n C_0({\mathcal G}amma)$ is a $C^*$-subalgebra of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$,
which is isomorphic to $C_0({\mathcal G}amma)$.
\begin{definition}\rm\label{omega}
Let $I$ be an ideal of the crossed product ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$.
For each $n\in\mathbb{N}$,
we define the closed subset $X_I^{(n)}$ of ${\mathcal G}amma$ by
$$X_I^{(n)}=\{\gamma\in{\mathcal G}amma\mid
f(\gamma)=0\mbox{ for all }f\in C_0({\mathcal G}amma)\mbox{ with }p_nf\in I\}.$$
Set $X_I=X_I^{(0)}$, $X_I^{(\infty)}=\bigcap_{n=1}^\infty X_I^{(n)}$,
and denote by $\widetilde{X}_I$
the pair $(X_I,X_I^{(\infty)})$ of subsets of ${\mathcal G}amma$.
\varepsilonnd{definition}
In other words, $X_I^{(n)}$ is determined by
$p_n C_0({\mathcal G}amma\setminus X_I^{(n)})=I\cap p_n C_0({\mathcal G}amma)$.
One can easily see that
$X_{I_1\cap I_2}^{(n)}=X_{I_1}^{(n)}\cup X_{I_2}^{(n)}$ for any $n\in\mathbb{N}$,
hence $X_{I_1\cap I_2}=X_{I_1}\cup X_{I_2},\
X_{I_1\cap I_2}^{(\infty)}=X_{I_1}^{(\infty)}\cup X_{I_2}^{(\infty)}$
and that $I_1\subset I_2$ implies
$X_{I_1}^{(n)}\supset X_{I_2}^{(n)}$ for any $n\in\mathbb{N}$, hence implies
$X_{I_1}\supset X_{I_2},\ X_{I_1}^{(\infty)}\supset X_{I_2}^{(\infty)}$.
For $n\in\mathbb{N}$, the set $X_I^{(n)}$ can be described
only in terms of $X_I$ and $X_I^{(\infty)}$.
\begin{lemma}\label{X_I^{(n)}}
For an ideal $I$ of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$, we have
\begin{align*}
X_I^{(n)}&=X_I^{(\infty)}\cup\bigcup_{i=n+1}^\infty (X_I+\omega_i),
\varepsilonnd{align*}
for any $n\in\mathbb{N}$.
\varepsilonnd{lemma}
\begin{proposition}f
Let $\gamma$ be an element of $X_I$
and $i$ be a positive integer grater than $n$.
Take $f\in C_0({\mathcal G}amma)$ with $p_nf\in I$.
Since
$$S_{i}^*p_nfS_{i}=S_{i}^*fS_{i}=S_{i}^*S_{i}\sigma_{\omega_{i}}f
=\sigma_{\omega_{i}}f,$$
we have $\sigma_{\omega_{i}}f\in I\cap C_0({\mathcal G}amma)$.
Since $\gamma\in X_I$, we have $\sigma_{\omega_{i}}f(\gamma)=0$.
Hence $f(\gamma+\omega_{i})=0$ for any $f\in C_0({\mathcal G}amma)$ with $p_nf\in I$.
It implies $\gamma+\omega_{i}\in X_I^{(n)}$.
Thus $X_I^{(n)}\supset X_I+\omega_i$ for any $i>n$.
For $n\leq m$, we have $X_I^{(n)}\supset X_I^{(m)}$ because $p_np_m=p_m$.
Therefore $X_I^{(n)}\supset X_I^{(\infty)}$.
Thus $X_I^{(n)}\supset
X_I^{(\infty)}\cup\bigcup_{i=n+1}^\infty (X_I+\omega_i)$.
Conversely, take
$\gamma\notin X_I^{(\infty)}\cup\bigcup_{i=n+1}^\infty (X_I+\omega_i)$.
Since $\gamma\notin X_I^{(\infty)}$, we can find a positive integer $m$ so that
$\gamma\notin X_I^{(m)}$.
When $m\leq n$, we see that $\gamma\notin X_I^{(n)}$.
We will show $\gamma\notin X_I^{(n)}$ in the case $m>n$.
Since $\gamma\notin X_I^{(m)}$,
there exists $f\in C_0({\mathcal G}amma)$ such that $p_mf\in I$ and $f(\gamma)\neq 0$.
For each $i=n+1,n+2,\ldots,m$, there exists $f_i\in C_0({\mathcal G}amma)\cap I$
such that $f_i(\gamma-\omega_i)\neq 0$ because $\gamma\notin X_I+\omega_i$.
Set $g=f\prod_{i=n+1}^m\sigma_{-\omega_i}f_i$.
We have $g(\gamma)\neq 0$ and
$$p_n g=p_m g+\sum_{i=n+1}^mS_iS_i^*g=p_m g+\sum_{i=n+1}^mS_i(\sigma_{\omega_i}g)S_i^*\in I.$$
Therefore $\gamma\notin X_I^{(n)}$.
Thus we have
$X_I^{(n)}=X_I^{(\infty)}\cup\bigcup_{i=n+1}^\infty (X_I+\omega_i)$.
\varepsilonnd{proposition}f
\begin{definition}\rm
A subset $X$ of ${\mathcal G}amma$ is called {\varepsilonm $\omega$-invariant}
if $X$ is a closed set with $X+\omega_i\subset X$ for any $i\in\mathbb{Z}_+$.
For an $\omega$-invariant set $X$, we define a closed set $H_X$ by
$$H_X=\overline{X\setminus\bigcup_{i=1}^\infty(X+\omega_i)}\ \cup\
\bigcap_{n=1}^\infty\overline{\bigcup_{i=n}^\infty(X+\omega_i)}.$$
\varepsilonnd{definition}
Note that $H_X$ is a closed subset of $X$.
\begin{definition}\rm
A pair $\widetilde{X}=(X,X^\infty)$ of subsets of ${\mathcal G}amma$ is called
{\varepsilonm $\omega$-invariant} if $X$ is an $\omega$-invariant set, and
$X^\infty$ is a closed set satisfying $H_X\subset X^\infty\subset X$.
\varepsilonnd{definition}
\begin{proposition}
For any ideal $I$ of the crossed product ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$,
the pair $\widetilde{X}_I$ is $\omega$-invariant.
\varepsilonnd{proposition}
\begin{proposition}f
By Lemma \ref{X_I^{(n)}},
we have $X_I=X_I^{(\infty)}\cup \bigcup_{i=1}^\infty(X_I+\omega_i)$.
From this, we see that $X_I$ is $\omega$-invariant and that
$X_I\setminus\bigcup_{i=1}^\infty(X_I+\omega_i)\subset
X_I^{(\infty)}\subset X_I$.
By Lemma \ref{X_I^{(n)}},
we have $\overline{\bigcup_{i=n}^\infty(X+\omega_i)}\subset\overline{X_I^{(n)}}
=X_I^{(n)}$.
Hence $\bigcap_{n=1}^\infty\overline{\bigcup_{i=n}^\infty(X+\omega_i)}\subset
\bigcap_{n=1}^\infty X_I^{(n)}=X_I^{(\infty)}$.
Therefore we get $H_X\subset X_I^{(\infty)}\subset X_I$.
\varepsilonnd{proposition}f
We will show that for an $\omega$-invariant pair $\widetilde{X}$,
there exists a gauge invariant ideal $I$ such that
$\widetilde{X}_I=\widetilde{X}$ (Proposition \ref{exist}).
\begin{lemma}\label{X^{(n)}}
Let $\widetilde{X}=(X,X^{(\infty)})$ be an $\omega$-invariant pair.
For $n\in\mathbb{N}$, set $X^{(n)}=X^{(\infty)}\cup\bigcup_{i=n+1}^\infty(X+\omega_i)$.
Then we have the following.
\benu
\item $X^{(n)}$ is closed for all $n\in\mathbb{N}$.
\item $X=X^{(0)}$, $X^{(\infty)}=\bigcap_{n=1}^\infty X^{(n)}$.
\item For $0\leq n<m$, $X^{(n)}=X^{(m)}\cup\bigcup_{i=n+1}^m(X+\omega_i)$.
\item For a positive integer $n$,
$$X=\bigcup_{\mu\in{\mathcal W}_n}(X^{(n)}+\omega_\mu)\cup
\bigcap_{k=1}^\infty \bigg( \bigcup_{\mu\in{\mathcal W}_n^{(k)}}(X+\omega_\mu)\bigg).$$
\varepsilonnd{enumerate}
\varepsilonnd{lemma}
\begin{proposition}f
\benu
\item Take $\gamma\in\overline{X^{(n)}}$ for a positive integer $n$.
If $U\cap X^{(\infty)}\neq\varepsilonmptyset$ for all neighborhood $U$ of $\gamma$,
then $\gamma\in X^{(\infty)}\subset X^{(n)}$ because $X^{(\infty)}$ is closed.
Otherwise, we can find a positive integer $i_U$ grater than $n$ with
$U\cap (X+\omega_{i_U})\neq\varepsilonmptyset$ for any neighborhood $U$ of $\gamma$.
If there exists $i$ such that $i_U=i$ eventually,
then $\gamma\in X+\omega_i\subset X^{(n)}$ because $X+\omega_i$ is closed.
If there are no such $i$, then we can see that
$\gamma\in\overline{\bigcup_{i=m}^\infty(X+\omega_i)}$ for any $m$ with $m>n$.
Hence $\gamma\in H_{X}\subset X^{(\infty)}\subset X^{(n)}$.
Thus we have proved that $\gamma\in X^{(n)}$,
from which it follows that $X^{(n)}$ is closed.
\item Since
$X\setminus\bigcup_{i=1}^\infty(X+\omega_i)\subset X^{(\infty)}\subset X$,
we have $X=X^{(0)}$.
We see that
\begin{align*}
\bigcap_{n=1}^\infty X^{(n)}
&=\bigcap_{n=1}^\infty
\bigg(X^{(\infty)}\cup\bigcup_{i=n+1}^\infty(X+\omega_i)\bigg)
=X^{(\infty)}\cup\bigcap_{n=1}^\infty
\bigg(\bigcup_{i=n+1}^\infty(X+\omega_i)\bigg).
\varepsilonnd{align*}
Since $\bigcap_{n=1}^\infty\left(\bigcup_{i=n+1}^\infty(X+\omega_i)\right)
\subset H_{X}\subset X^{(\infty)}$,
we have $\bigcap_{n=1}^\infty X^{(n)}=X^{(\infty)}$.
\item It is obvious by the definition.
\item For a positive integer $n$, we have
$X=X^{(n)}\cup\bigcup_{i=1}^n(X+\omega_i)$ by (iii).
Recursively, we get
$X=\bigcup_{m=0}^{k-1}\big(\bigcup_{\mu\in{\mathcal W}_n^{(m)}}(X^{(n)}+\omega_\mu)\big)
\cup\bigcup_{\mu\in{\mathcal W}_n^{(k)}}(X+\omega_\mu)$ for any positive integer $k$.
Hence $X=\bigcup_{\mu\in{\mathcal W}_n}(X^{(n)}+\omega_\mu)\cup
\bigcap_{k=1}^\infty \big( \bigcup_{\mu\in{\mathcal W}_n^{(k)}}(X+\omega_\mu)\big)$.
\varepsilonnd{proposition}f
\varepsilonnd{enumerate}
\begin{definition}\rm
For an $\omega$-invariant pair $\widetilde{X}=(X,X^{(\infty)})$,
we define $I_{\widetilde{X}}\subset{\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ by
$$I_{\widetilde{X}}=\cspa\{S_\mu p_n fS_\nu^*\mid
\mu,\nu\in{\mathcal W}_\infty,\ f\in C_0({\mathcal G}amma\setminus X^{(n)}),\ n\in\mathbb{N}\},$$
where $X^{(n)}=X^{(\infty)}\cup\bigcup_{i=n+1}^\infty(X+\omega_i)$.
\varepsilonnd{definition}
\begin{proposition}\label{I_X,Xinfty}
For an $\omega$-invariant pair $\widetilde{X}=(X,X^{(\infty)})$,
the set $I_{\widetilde{X}}$ becomes a gauge invariant ideal of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$.
\varepsilonnd{proposition}
| 3,572 | 50,987 |
en
|
train
|
0.36.2
|
We will show that for an $\omega$-invariant pair $\widetilde{X}$,
there exists a gauge invariant ideal $I$ such that
$\widetilde{X}_I=\widetilde{X}$ (Proposition \ref{exist}).
\begin{lemma}\label{X^{(n)}}
Let $\widetilde{X}=(X,X^{(\infty)})$ be an $\omega$-invariant pair.
For $n\in\mathbb{N}$, set $X^{(n)}=X^{(\infty)}\cup\bigcup_{i=n+1}^\infty(X+\omega_i)$.
Then we have the following.
\benu
\item $X^{(n)}$ is closed for all $n\in\mathbb{N}$.
\item $X=X^{(0)}$, $X^{(\infty)}=\bigcap_{n=1}^\infty X^{(n)}$.
\item For $0\leq n<m$, $X^{(n)}=X^{(m)}\cup\bigcup_{i=n+1}^m(X+\omega_i)$.
\item For a positive integer $n$,
$$X=\bigcup_{\mu\in{\mathcal W}_n}(X^{(n)}+\omega_\mu)\cup
\bigcap_{k=1}^\infty \bigg( \bigcup_{\mu\in{\mathcal W}_n^{(k)}}(X+\omega_\mu)\bigg).$$
\varepsilonnd{enumerate}
\varepsilonnd{lemma}
\begin{proposition}f
\benu
\item Take $\gamma\in\overline{X^{(n)}}$ for a positive integer $n$.
If $U\cap X^{(\infty)}\neq\varepsilonmptyset$ for all neighborhood $U$ of $\gamma$,
then $\gamma\in X^{(\infty)}\subset X^{(n)}$ because $X^{(\infty)}$ is closed.
Otherwise, we can find a positive integer $i_U$ grater than $n$ with
$U\cap (X+\omega_{i_U})\neq\varepsilonmptyset$ for any neighborhood $U$ of $\gamma$.
If there exists $i$ such that $i_U=i$ eventually,
then $\gamma\in X+\omega_i\subset X^{(n)}$ because $X+\omega_i$ is closed.
If there are no such $i$, then we can see that
$\gamma\in\overline{\bigcup_{i=m}^\infty(X+\omega_i)}$ for any $m$ with $m>n$.
Hence $\gamma\in H_{X}\subset X^{(\infty)}\subset X^{(n)}$.
Thus we have proved that $\gamma\in X^{(n)}$,
from which it follows that $X^{(n)}$ is closed.
\item Since
$X\setminus\bigcup_{i=1}^\infty(X+\omega_i)\subset X^{(\infty)}\subset X$,
we have $X=X^{(0)}$.
We see that
\begin{align*}
\bigcap_{n=1}^\infty X^{(n)}
&=\bigcap_{n=1}^\infty
\bigg(X^{(\infty)}\cup\bigcup_{i=n+1}^\infty(X+\omega_i)\bigg)
=X^{(\infty)}\cup\bigcap_{n=1}^\infty
\bigg(\bigcup_{i=n+1}^\infty(X+\omega_i)\bigg).
\varepsilonnd{align*}
Since $\bigcap_{n=1}^\infty\left(\bigcup_{i=n+1}^\infty(X+\omega_i)\right)
\subset H_{X}\subset X^{(\infty)}$,
we have $\bigcap_{n=1}^\infty X^{(n)}=X^{(\infty)}$.
\item It is obvious by the definition.
\item For a positive integer $n$, we have
$X=X^{(n)}\cup\bigcup_{i=1}^n(X+\omega_i)$ by (iii).
Recursively, we get
$X=\bigcup_{m=0}^{k-1}\big(\bigcup_{\mu\in{\mathcal W}_n^{(m)}}(X^{(n)}+\omega_\mu)\big)
\cup\bigcup_{\mu\in{\mathcal W}_n^{(k)}}(X+\omega_\mu)$ for any positive integer $k$.
Hence $X=\bigcup_{\mu\in{\mathcal W}_n}(X^{(n)}+\omega_\mu)\cup
\bigcap_{k=1}^\infty \big( \bigcup_{\mu\in{\mathcal W}_n^{(k)}}(X+\omega_\mu)\big)$.
\varepsilonnd{proposition}f
\varepsilonnd{enumerate}
\begin{definition}\rm
For an $\omega$-invariant pair $\widetilde{X}=(X,X^{(\infty)})$,
we define $I_{\widetilde{X}}\subset{\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ by
$$I_{\widetilde{X}}=\cspa\{S_\mu p_n fS_\nu^*\mid
\mu,\nu\in{\mathcal W}_\infty,\ f\in C_0({\mathcal G}amma\setminus X^{(n)}),\ n\in\mathbb{N}\},$$
where $X^{(n)}=X^{(\infty)}\cup\bigcup_{i=n+1}^\infty(X+\omega_i)$.
\varepsilonnd{definition}
\begin{proposition}\label{I_X,Xinfty}
For an $\omega$-invariant pair $\widetilde{X}=(X,X^{(\infty)})$,
the set $I_{\widetilde{X}}$ becomes a gauge invariant ideal of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$.
\varepsilonnd{proposition}
\begin{proposition}f
Clearly $I_{\widetilde{X}}$ is a $*$-invariant closed linear space,
and is invariant under the gauge action $\beta$ because
$\beta_t(S_\mu p_n fS_\nu^*)=t^{|\mu|-|\nu|}S_\mu p_n fS_\nu^*$
for $t\in\mathbb{T}$.
To prove that $I_{\widetilde{X}}$ is an ideal,
it suffices to show that for any $\mu_1,\nu_1,\mu_2,\nu_2\in{\mathcal W}_\infty$
and any $f\in C_0({\mathcal G}amma\setminus X^{(n)}),\ g\in C_0({\mathcal G}amma)$,
the product $xy$ of $x=S_{\mu_1}p_n fS_{\nu_1}^*\in I_{\widetilde{X}}$
and $y=S_{\mu_2}gS_{\nu_2}^*\in{\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ is in $I_{\widetilde{X}}$.
If $S_{\nu_1}^*S_{\mu_2}=0$ or $S_{\nu_1}^*S_{\mu_2}=S_\mu^*$
for some $\mu\in{\mathcal W}_\infty$,
then it is easy to see that $xy\in I_{\widetilde{X}}$.
Otherwise $S_{\nu_1}^*S_{\mu_2}=S_\mu$
for some $\mu=(i_1,i_2,\ldots,i_k)\in{\mathcal W}_\infty$ with $\mu\neq\varepsilonmptyset$.
When $i_1\leq n$, we have $p_n fS_\mu=p_n S_\mu\sigma_{\omega_\mu}f=0$.
Hence $xy=0\in I_{\widetilde{X}}$.
When $i_1> n$, we have
$p_n fS_\mu=p_n S_\mu\sigma_{\omega_\mu}f=S_\mu\sigma_{\omega_\mu}f$.
Now, $f\in C_0({\mathcal G}amma\setminus X^{(n)})$ implies
$\sigma_{\omega_\mu}f\in C_0({\mathcal G}amma\setminus X)$ because
$X+\omega_\mu\subset X+\omega_{i_1}\subset X^{(n)}$.
Hence we have $xy\in I_{\widetilde{X}}$.
It completes the proof.
\varepsilonnd{proposition}f
\begin{proposition}\label{exist}
Let $\widetilde{X}=(X,X^{(\infty)})$ be an $\omega$-invariant pair,
and set $I=I_{\widetilde{X}}$.
Then $\widetilde{X}_I=\widetilde{X}$.
\varepsilonnd{proposition}
\begin{proposition}f
By the definition of $I$, we get $X_I^{(n)}\subset X^{(n)}$ for any $n\in\mathbb{N}$.
We will first prove that $X_I=X$.
To the contrary, assume that $X_I\subsetneqq X$.
Then there exists $f\in I\cap C_0({\mathcal G}amma)$
such that $f(\gamma_0)=1$ for some $\gamma_0\in X$.
Since $f\in I$, there exist $n_l\in\mathbb{N}$,
$f_l\in C_0({\mathcal G}amma\setminus X^{(n_l)})$ and
$\mu_l,\nu_l\in{\mathcal W}_\infty\ (l=1,2,\ldots,L)$ such that
$$\bigg\| f-\sum_{l=1}^LS_{\mu_l}p_{n_l}f_lS_{\nu_l}^*\bigg\|<\frac12.$$
Take a positive integer $n$ so large that
$n_l\leq n$ and $\mu_l,\nu_l\in{\mathcal W}_n$ for $l=1,2,\ldots,L$.
For any $\mu_0\in{\mathcal W}_n$,
we have $p_nS_{\mu_0}^*fS_{\mu_0}p_n=p_n\sigma_{\omega_{\mu_0}}f$
and $\sigma_{\omega_{\mu_0}}f(\gamma_0-\omega_{\mu_0})=1$.
For $l$ with $\mu_l=\nu_l=\mu_0$, we have
$p_nS_{\mu_0}^*(S_{\mu_l}p_{n_l}f_lS_{\nu_l}^*)S_{\mu_0} p_n=p_n f_l$.
For $l$ with $\mu_l\nu=\nu_l\nu=\mu_0$ for some
$\nu=(i_1,i_2,\ldots,i_k)\in{\mathcal W}_n$ with $i_1>n_l$,
we have
$p_nS_{\mu_0}^*(S_{\mu_l}p_{n_l}f_lS_{\nu_l}^*)S_{\mu_0} p_n
=p_n\sigma_{\omega_{\nu}}f_l$.
We have $\sigma_{\omega_{\nu}}f_l\in C_0({\mathcal G}amma\setminus X)$,
because $X+\omega_{\nu}\subset X+\omega_{i_1}\subset X^{(n_l)}$.
For other $l$, we have
$p_nS_{\mu_0}^*(S_{\mu_l}p_{n_l}f_lS_{\nu_l}^*)S_{\mu_0} p_n=0$.
Hence we get
$$\bigg\|\sigma_{\omega_{\mu_0}}f-\sum_{l=1}^L g_l\bigg\|=
\bigg\|p_n\bigg(\sigma_{\omega_{\mu_0}}f-\sum_{l=1}^L g_l\bigg)\bigg\|=\bigg\|p_nS_{\mu_0}^*\bigg(f-\sum_{l=1}^LS_{\mu_l}p_{n_l}f_lS_{\nu_l}^*\bigg)
S_{\mu_0}p_n\bigg\|<\frac12,$$
where $g_l\in C_0({\mathcal G}amma\setminus X^{(n_l)})$ when $\mu_l=\nu_l=\mu_0$,
and $g_l\in C_0({\mathcal G}amma\setminus X)$ when $\mu_l\nu=\nu_l\nu=\mu_0$ for some
$\nu=(i_1,i_2,\ldots,i_k)\in{\mathcal W}_n$ with $i_1>n_l$, and $g_l=0$ otherwise.
To derive a contradiction, it suffices to find $\mu_0\in{\mathcal W}_n$ such that
$g_l(\gamma_0-\omega_{\mu_0})=0$ for any $l$.
By Lemma \ref{X^{(n)}} (iv), we have either $\gamma_0\in \bigcap_{m=1}^\infty
\big( \bigcup_{\mu\in{\mathcal W}_n^{(m)}}(X+\omega_\mu)\big)$
or $\gamma_0\in X^{(n)}+\omega_\mu$ for some $\mu\in{\mathcal W}_n$.
When $\gamma_0\in \bigcap_{m=1}^\infty
\big( \bigcup_{\mu\in{\mathcal W}_n^{(m)}}(X+\omega_\mu)\big)$,
take $\mu_0\in{\mathcal W}_n$ so that $|\mu_0|>|\mu_l|,|\nu_l|$ for $l=1,2,\ldots,L$
and $\gamma_0\in X+\omega_{\mu_0}$.
Then $\mu_l=\nu_l=\mu_0$ never occurs.
Hence $g_l\in C_0({\mathcal G}amma\setminus X)$ for any $l$.
We get $g_l(\gamma_0-\omega_{\mu_0})=0$ because $\gamma_0-\omega_{\mu_0}\in X$.
When $\gamma_0\in X^{(n)}+\omega_\mu$ for some $\mu\in{\mathcal W}_n$, take $\mu_0=\mu$.
Since $\gamma_0-\omega_{\mu_0}\in X^{(n)}\subset X^{(n_l)}\subset X$,
we have $g_l(\gamma_0-\omega_{\mu_0})=0$
either if $g_l\in C_0({\mathcal G}amma\setminus X^{(n_l)})$ or
if $g_l\in C_0({\mathcal G}amma\setminus X)$.
Hence $g_l(\gamma_0-\omega_{\mu_0})=0$ for any $l$.
Therefore we have $X_I=X$.
Next we will show that $X_I^{(n)}=X^{(n)}$ for a positive integer $n$.
To derive a contradiction, assume that $X_I^{(n)}\subsetneqq X^{(n)}$.
Then there exists $f\in C_0({\mathcal G}amma)$
such that $p_nf\in I$ and $f(\gamma_0)=1$ for some $\gamma_0\in X^{(n)}$.
Since $p_nf\in I$, there exist $n_l\in\mathbb{N}$,
$f_l\in C_0({\mathcal G}amma\setminus X^{(n_l)})$ and
$\mu_l,\nu_l\in{\mathcal W}_\infty\ (l=1,2,\ldots,L)$ such that
$$\bigg\| p_nf-\sum_{l=1}^LS_{\mu_l}p_{n_l}f_lS_{\nu_l}^*\bigg\|<\frac12.$$
Take a positive integer $m$ so large that $\mu_l,\nu_l\in{\mathcal W}_{m}$, $n_l\leq m$
for $l=1,2,\ldots,L$ and $n\leq m$.
By Lemma \ref{X^{(n)}} (iii), we have
$X^{(n)}=X^{(m)}\cup\bigcup_{i=n+1}^{m}(X+\omega_i)$.
When $\gamma_0\in X^{(m)}$, we have $f_l(\gamma_0)=0$ for any $l$.
On the other hand, we get $\|f-\sum_{\mu_l=\nu_l=\varepsilonmptyset}f_l\|<1/2$
because
$$p_{m}\bigg(p_nf-\sum_{l=1}^LS_{\mu_l}p_{n_l}f_lS_{\nu_l}^*
\bigg)p_{m}=p_{m}f-\sum_{\mu_l=\nu_l=\varepsilonmptyset}p_{m}f_l.$$
This is a contradiction.
When $\gamma_0\in X+\omega_i$ for some $i$ with $n<i\leq m$,
we have $\sigma_{\omega_i}f=S_i^*(p_nf)S_i\in I$ and
$\sigma_{\omega_i}f(\gamma_0-\omega_i)=1$.
This contradicts the fact that $X_I=X$.
Therefore $X_I^{(n)}=X^{(n)}$ for a positive integer $n$.
Hence $X_I^{(\infty)}=\bigcap_{n=1}^\infty X_I^{(n)}=
\bigcap_{n=1}^\infty X^{(n)}=X^{(\infty)}$.
We have shown that $\widetilde{X}_I=\widetilde{X}$.
\varepsilonnd{proposition}f
| 4,082 | 50,987 |
en
|
train
|
0.36.3
|
When $\gamma_0\in \bigcap_{m=1}^\infty
\big( \bigcup_{\mu\in{\mathcal W}_n^{(m)}}(X+\omega_\mu)\big)$,
take $\mu_0\in{\mathcal W}_n$ so that $|\mu_0|>|\mu_l|,|\nu_l|$ for $l=1,2,\ldots,L$
and $\gamma_0\in X+\omega_{\mu_0}$.
Then $\mu_l=\nu_l=\mu_0$ never occurs.
Hence $g_l\in C_0({\mathcal G}amma\setminus X)$ for any $l$.
We get $g_l(\gamma_0-\omega_{\mu_0})=0$ because $\gamma_0-\omega_{\mu_0}\in X$.
When $\gamma_0\in X^{(n)}+\omega_\mu$ for some $\mu\in{\mathcal W}_n$, take $\mu_0=\mu$.
Since $\gamma_0-\omega_{\mu_0}\in X^{(n)}\subset X^{(n_l)}\subset X$,
we have $g_l(\gamma_0-\omega_{\mu_0})=0$
either if $g_l\in C_0({\mathcal G}amma\setminus X^{(n_l)})$ or
if $g_l\in C_0({\mathcal G}amma\setminus X)$.
Hence $g_l(\gamma_0-\omega_{\mu_0})=0$ for any $l$.
Therefore we have $X_I=X$.
Next we will show that $X_I^{(n)}=X^{(n)}$ for a positive integer $n$.
To derive a contradiction, assume that $X_I^{(n)}\subsetneqq X^{(n)}$.
Then there exists $f\in C_0({\mathcal G}amma)$
such that $p_nf\in I$ and $f(\gamma_0)=1$ for some $\gamma_0\in X^{(n)}$.
Since $p_nf\in I$, there exist $n_l\in\mathbb{N}$,
$f_l\in C_0({\mathcal G}amma\setminus X^{(n_l)})$ and
$\mu_l,\nu_l\in{\mathcal W}_\infty\ (l=1,2,\ldots,L)$ such that
$$\bigg\| p_nf-\sum_{l=1}^LS_{\mu_l}p_{n_l}f_lS_{\nu_l}^*\bigg\|<\frac12.$$
Take a positive integer $m$ so large that $\mu_l,\nu_l\in{\mathcal W}_{m}$, $n_l\leq m$
for $l=1,2,\ldots,L$ and $n\leq m$.
By Lemma \ref{X^{(n)}} (iii), we have
$X^{(n)}=X^{(m)}\cup\bigcup_{i=n+1}^{m}(X+\omega_i)$.
When $\gamma_0\in X^{(m)}$, we have $f_l(\gamma_0)=0$ for any $l$.
On the other hand, we get $\|f-\sum_{\mu_l=\nu_l=\varepsilonmptyset}f_l\|<1/2$
because
$$p_{m}\bigg(p_nf-\sum_{l=1}^LS_{\mu_l}p_{n_l}f_lS_{\nu_l}^*
\bigg)p_{m}=p_{m}f-\sum_{\mu_l=\nu_l=\varepsilonmptyset}p_{m}f_l.$$
This is a contradiction.
When $\gamma_0\in X+\omega_i$ for some $i$ with $n<i\leq m$,
we have $\sigma_{\omega_i}f=S_i^*(p_nf)S_i\in I$ and
$\sigma_{\omega_i}f(\gamma_0-\omega_i)=1$.
This contradicts the fact that $X_I=X$.
Therefore $X_I^{(n)}=X^{(n)}$ for a positive integer $n$.
Hence $X_I^{(\infty)}=\bigcap_{n=1}^\infty X_I^{(n)}=
\bigcap_{n=1}^\infty X^{(n)}=X^{(\infty)}$.
We have shown that $\widetilde{X}_I=\widetilde{X}$.
\varepsilonnd{proposition}f
By Proposition \ref{exist}, the map $I\mapsto \widetilde{X}_I$
from the set of gauge invariant ideals $I$ of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$
to the set of $\omega$-invariant pairs is surjective.
Now, we turn to showing that this map is injective (Proposition \ref{unique}).
To do so, we investigate the quotient $({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I$ of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ by an ideal $I$
which is not ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$.
Since $I\cap C_0({\mathcal G}amma)=C_0({\mathcal G}amma\setminus X_I)$, a $C^*$-subalgebra
$C_0({\mathcal G}amma)/(I\cap C_0({\mathcal G}amma))$ of $({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I$ is isomorphic to $C_0(X_I)$.
We will consider $C_0(X_I)$ as a $C^*$-subalgebra of $({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I$.
We will use the same symbols
$S_1,S_2,\ldots\in M(({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I)$ as the ones in $M({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)$ for denoting
the isometries of ${\mathcal O}_\infty$ which is naturally embedded into $M(({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I)$.
For an $\omega$-invariant set $X$,
we can define a $*$-homomorphism $\sigma_{\omega_\mu}:C_0(X)\to C_0(X)$
for $\mu\in{\mathcal W}_\infty$.
This map $\sigma_{\omega_\mu}$ is always surjective,
but it is injective only in the case that $X\subset X+\omega_\mu$,
which is equivalent to $X=X+\omega_\mu$.
One can easily verify the following.
\begin{lemma}\label{cp/I}
Let $I$ be an ideal that is not ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$.
For $\mu,\nu\in{\mathcal W}_\infty$ and $f\in C_0(X_I)\subset({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I$,
the following hold.
\benu
\item $S_\mu fS_\nu^*=0$ if and only if $f=0$.
\item For $n\in\mathbb{N}$, $p_nf=0$ if and only if $f\in C_0(X_I\setminus X_I^{(n)})$.
\item $fS_\mu=S_\mu \sigma_{\omega_\mu}f$.
\item $({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I=\cspa\{S_\mu fS_\nu^*\mid\mu,\nu\in{\mathcal W}_\infty,\ f\in C_0(X_I)\}$.
\varepsilonnd{enumerate}
\varepsilonnd{lemma}
We define a $C^*$-subalgebra of $({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I$,
which corresponds to the AF-core for Cuntz algebras.
\begin{definition}\rm\label{AFcore}
Let $I$ be an ideal that is not ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$.
We define $C^*$-subalgebras of $({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I$ by
\begin{align*}
{\mathcal G}_I^{(n,k)}&=\spa\{S_\mu fS_\nu^*\mid \mu,\nu\in{\mathcal W}_n^{(k)},\
f\in C_0(X_I)\},\\
{\mathcal F}_I^{(n,k)}&=\spa\{S_\mu p_nfS_\nu^*\mid \mu,\nu\in{\mathcal W}_n^{(k)},\
f\in C_0(X_I)\},\\
{\mathcal F}_I^{(n)}&=\spa\{S_\mu fS_\nu^*\mid \mu,\nu\in{\mathcal W}_n,\
0\leq |\mu|=|\nu|\leq n,\ f\in C_0(X_I)\},\\
{\mathcal F}_I&=\cspa\{S_\mu fS_\nu^*\mid \mu,\nu\in{\mathcal W}_\infty,\
|\mu|=|\nu|,\ f\in C_0(X_I)\},
\varepsilonnd{align*}
for $n\in\mathbb{Z}_+, 0\leq k\leq n$.
\varepsilonnd{definition}
\begin{lemma}\label{F_n}
Let $I$ be an ideal that is not ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$.
For $n\in\mathbb{Z}_+, 0\leq k\leq n$, we have the following.
\benu
\item ${\mathcal G}_I^{(n,k)}\cong \mathbb{M}_{n^k}\otimes C_0(X_I)$.
\item ${\mathcal F}_I^{(n,k)}\cong \mathbb{M}_{n^k}\otimes C_0(X_I^{(n)})$.
\item ${\mathcal F}_I^{(n)}\cong\bigoplus_{k=0}^{n-1}{\mathcal F}_I^{(n,k)}\oplus {\mathcal G}_I^{(n,n)}$.
\item $\bigcup_{n=1}^\infty{\mathcal F}_I^{(n)}$ is dense in ${\mathcal F}_I$.
\varepsilonnd{enumerate}
\varepsilonnd{lemma}
\begin{proposition}f
\benu
\item Since the set ${\mathcal W}_n^{(k)}$ has $n^k$ elements, we may use
$\{e_{\mu,\nu}\}_{\mu,\nu\in{\mathcal W}_n^{(k)}}$ for denoting the matrix units of
$\mathbb{M}_{n^k}$.
One can easily see that
$$\mathbb{M}_{n^k}\otimes C_0(X_I)\ni e_{\mu,\nu}\otimes f
\mapsto S_\mu fS_\nu^*\in {\mathcal G}_I^{(n,k)}$$
gives us an isomorphism from $\mathbb{M}_{n^k}\otimes C_0(X_I)$ to ${\mathcal G}_I^{(n,k)}$.
\item We can define a surjective map from ${\mathcal G}_I^{(n,k)}$ to ${\mathcal F}_I^{(n,k)}$ by
$${\mathcal G}_I^{(n,k)}\ni S_\mu fS_\nu^*\mapsto S_\mu p_n fS_\nu^*\in {\mathcal F}_I^{(n,k)}.$$
Its kernel is $\mathbb{M}_{n^k}\otimes C_0(X_I\setminus X_I^{(n)})$
under the isomorphism ${\mathcal G}_I^{(n,k)}\cong \mathbb{M}_{n^k}\otimes C_0(X_I)$
by Lemma \ref{cp/I} (ii).
Hence we have ${\mathcal F}_I^{(n,k)}\cong \mathbb{M}_{n^k}\otimes C_0(X_I^{(n)})$.
\item It can be done just by computation.
\item Obvious by the definitions of ${\mathcal F}_I^{(n)}$ and ${\mathcal F}_I$.
\varepsilonnd{proposition}f
\varepsilonnd{enumerate}
We will often identify ${\mathcal G}_I^{(n,n)}$ with $C_0(X_I,\mathbb{M}_{n^n})$.
The following lemma essentially appeared in \cite{C}.
\begin{lemma}\label{isom}
For $i=1,2$, let $E_i$ be a conditional expectation
from a $C^*$-algebra $A_i$ onto a $C^*$-subalgebra $B_i$ of $A_i$.
Let $\varphi:A_1\to A_2$ be a $*$-homomorphism
with $\varphi\circ E_1=E_2\circ\varphi$.
If the restriction of $\varphi$ on $B_1$ is injective and $E_1$ is faithful,
then $\varphi$ is injective.
\varepsilonnd{lemma}
For an ideal $I$ which is invariant under the gauge action $\beta$,
we can extend the gauge action on ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ to one on $({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I$,
which is also denoted by $\beta$.
The following lemma is standard.
\begin{lemma}\label{cond.exp1}
Let $I$ be a gauge invariant ideal that is not ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$.
Then,
$$E_{I}:({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I\ni x\mapsto\int_{\mathbb{T}}\beta_t(x)dt\in ({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I$$
is a faithful conditional expectation onto ${\mathcal F}_{I}$,
where $dt$ is the normalized Haar measure on $\mathbb{T}$.
\varepsilonnd{lemma}
\begin{proposition}\label{unique}
For any gauge invariant ideal $I$, we have $I_{\widetilde{X}_I}=I$.
\varepsilonnd{proposition}
\begin{proposition}f
When $I={\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$, we have $X_I=X_I^{(\infty)}=\varepsilonmptyset$.
Thus $I_{\widetilde{X}_I}={\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$.
Let $I$ be a gauge invariant ideal that is not ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$
and set $J=I_{\widetilde{X}_I}$.
By the definition, $J\subset I$.
Hence there exists a surjective $*$-homomorphism $\pi:({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/J\to({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I$.
By Proposition \ref{exist} and Lemma \ref{F_n},
the restriction of $\pi$ on ${\mathcal F}_{J}^{(k)}$
is an isomorphism from ${\mathcal F}_{J}^{(k)}$ onto ${\mathcal F}_{I}^{(k)}$
and so the restriction of $\pi$ on ${\mathcal F}_{J}$ is an isomorphism
from ${\mathcal F}_{J}$ onto ${\mathcal F}_{I}$.
By Lemma \ref{cond.exp1}, there are faithful conditional expectations
$E_J:({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/J\to {\mathcal F}_{J}$ and $E_I:({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I\to{\mathcal F}_{I}$
with $E_I\circ \pi=\pi\circ E_{J}$.
By Lemma \ref{isom}, $\pi$ is injective.
Therefore $I_{\widetilde{X}_I}=I$.
\varepsilonnd{proposition}f
\begin{theorem}\label{OneToOne}
The maps $I\mapsto \widetilde{X}_I$ and
$\widetilde{X}\mapsto I_{\widetilde{X}}$ induce
a one-to-one correspondence
between the set of gauge invariant ideals of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$
and the set of $\omega$-invariant pairs of subsets of ${\mathcal G}amma$.
\varepsilonnd{theorem}
\begin{proposition}f
Combine Proposition \ref{exist} and Proposition \ref{unique}.
\varepsilonnd{proposition}f
| 3,877 | 50,987 |
en
|
train
|
0.36.4
|
\section{Primeness for $\omega$-invariant pairs}
In this section, we give a necessary condition for an ideal to be primitive
in terms of $\omega$-invariant pairs.
We will use it after in order to determine all primitive ideals.
An ideal of a $C^*$-algebra is called primitive
if it is a kernel of some irreducible representation.
A $C^*$-algebra is called primitive if $0$ is a primitive ideal.
When a $C^*$-algebra $A$ is separable, an ideal $I$ of $A$ is primitive
if and only if $I$ is prime, i.e.
for two ideals $I_1,I_2$ of $A$,
$I_1\cap I_2\subset I$ implies either $I_1\subset I$ or $I_2\subset I$.
We define primeness for $\omega$-invariant pairs.
For two $\omega$-invariant pair $\widetilde{X}_1=(X_1,X_1^{(\infty)}),\
\widetilde{X}_2=(X_2,X_2^{(\infty)})$,
we write $\widetilde{X}_1\subset \widetilde{X}_2$
if $X_1\subset X_2, X_1^{(\infty)}\subset X_2^{(\infty)}$
and denote by $\widetilde{X}_1\cup\widetilde{X}_2$
the $\omega$-invariant pair $(X_1\cup X_2,X_1^{(\infty)}\cup X_2^{(\infty)})$.
\begin{definition}\rm
An $\omega$-invariant pair $\widetilde{X}$ is called {\varepsilonm prime}
if $\widetilde{X}_1\cup \widetilde{X}_2\supset\widetilde{X}$
implies either $\widetilde{X}_1\supset\widetilde{X}$
or $\widetilde{X}_2\supset\widetilde{X}$
for two $\omega$-invariant pairs
$\widetilde{X}_1,\ \widetilde{X}_2$.
\varepsilonnd{definition}
\begin{proposition}\label{prime}
If an ideal $I$ of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ is primitive,
then $\widetilde{X}_I$ is a prime $\omega$-invariant pair.
\varepsilonnd{proposition}
\begin{proposition}f
Let $I$ be a primitive ideal of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$.
Take two $\omega$-invariant pairs
$\widetilde{X}_1$, $\widetilde{X}_2$ with
$\widetilde{X}_1\cup \widetilde{X}_2\supset \widetilde{X}_I$.
Set $I_1=I_{\widetilde{X}_1}$ and $I_2=I_{\widetilde{X}_2}$.
Then
$$I_1\cap I_2=I_{\widetilde{X}_1\cup \widetilde{X}_2}\subset
I_{\widetilde{X}_I}\subset I.$$
Since $I$ is prime, we have either $I_1\subset I$ or $I_2\subset I$.
Hence we get either $\widetilde{X}_1\supset \widetilde{X}_I$
or $\widetilde{X}_2\supset\widetilde{X}_I$.
Thus $\widetilde{X}_I$ is prime.
\varepsilonnd{proposition}f
In general, the converse of Proposition \ref{prime} is not true
(see Corollary \ref{primitive1} and Proposition \ref{primitive2}).
The ideal $I$ is prime if and only if
the equality $I_1\cap I_2=I$ implies either $I_1=I$ or $I_2=I$
for two ideals $I_1,I_2$
(see the proof of (iii)$\mathbb{R}ightarrow$(iv) of Proposition \ref{primepair}).
The following is the counterpart of this fact
for prime $\omega$-invariant pairs.
\begin{proposition}\label{primepair}
For an $\omega$-invariant pair $\widetilde{X}$,
the following are equivalent.
\benu
\item $\widetilde{X}$ is prime.
\item
For two $\omega$-invariant pairs $\widetilde{X}_1$,
$\widetilde{X}_2$,
the equality $\widetilde{X}_1\cup \widetilde{X}_2=\widetilde{X}$
implies either $\widetilde{X}_1=\widetilde{X}$
or $\widetilde{X}_2=\widetilde{X}$.
\item For two gauge invariant ideals $I_1,I_2$ of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$,
the equality $I_1\cap I_2=I_{\widetilde{X}}$ implies either
$I_1=I_{\widetilde{X}}$ or $I_2=I_{\widetilde{X}}$.
\item For two gauge invariant ideals $I_1,I_2$ of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$,
the inclusion $I_1\cap I_2\subset I_{\widetilde{X}}$ implies
either $I_1\subset I_{\widetilde{X}}$ or $I_2\subset I_{\widetilde{X}}$.
\varepsilonnd{enumerate}
\varepsilonnd{proposition}
\begin{proposition}f
(i)$\mathbb{R}ightarrow$(ii):
Take two $\omega$-invariant pairs $\widetilde{X}_1$,
$\widetilde{X}_2$ with $\widetilde{X}_1\cup \widetilde{X}_2=\widetilde{X}$.
By (i), we have either $\widetilde{X}_1\supset \widetilde{X}$
or $\widetilde{X}_2\supset \widetilde{X}$.
Hence we get either $\widetilde{X}_1=\widetilde{X}$
or $\widetilde{X}_2=\widetilde{X}$.
(ii)$\mathbb{R}ightarrow$(iii):
Take two gauge invariant ideals $I_1,I_2$
with $I_1\cap I_2=I_{\widetilde{X}}$.
We have $\widetilde{X}_{I_1}\cup \widetilde{X}_{I_2}=\widetilde{X}$.
By (ii), we have either $\widetilde{X}_{I_1}=\widetilde{X}$ or
$\widetilde{X}_{I_2}=\widetilde{X}$.
By Proposition \ref{unique},
we have either $I_1=I_{\widetilde{X}}$ or $I_2=I_{\widetilde{X}}$.
(iii)$\mathbb{R}ightarrow$(iv):
Take two gauge invariant ideals $I_1,I_2$
with $I_1\cap I_2\subset I_{\widetilde{X}}$.
Then we have
$$(I_1+I_{\widetilde{X}})\cap (I_2+I_{\widetilde{X}})=
(I_1\cap I_2) +I_{\widetilde{X}}=I_{\widetilde{X}}.$$
By (iii), either $I_1+I_{\widetilde{X}}=I_{\widetilde{X}}$ or
$I_2+I_{\widetilde{X}}=I_{\widetilde{X}}$ holds.
Hence we get
either $I_1\subset I_{\widetilde{X}}$ or $I_2\subset I_{\widetilde{X}}$.
(iv)$\mathbb{R}ightarrow$(i):
Similarly as the proof of Proposition \ref{prime}.
\varepsilonnd{proposition}f
We will use the implication (ii)$\mathbb{R}ightarrow$(i)
to determine which $\omega$-invariant pair is prime.
We also need a notion of primeness for $\omega$-invariant sets.
\begin{definition}\rm
An $\omega$-invariant set $X$ is called {\varepsilonm prime}
if $X_1\cup X_2\supset X$ implies either $X_1\supset X$ or $X_2\supset X$,
for any $\omega$-invariant sets $X_1,X_2$.
\varepsilonnd{definition}
We set ${\rm{sg}}(\omega)=\{\omega_\mu\mid \mu\in{\mathcal W}_\infty\}$
which is the semigroup generated by $\omega_1,\omega_2,\ldots$ and
denote by $\overline{\rm{sg}}(\omega)$ its closure.
Note that a closed subset $X$ of ${\mathcal G}amma$ is $\omega$-invariant
if and only if $X+\overline{\rm{sg}}(\omega)=X$.
For any $\gamma\in{\mathcal G}amma$, it is easy to see that
the set $\gamma+\overline{\rm{sg}}(\omega)$ is a prime $\omega$-invariant set.
The following is a necessary and sufficient condition
for an $\omega$-invariant set to be prime,
which can be considered as an analogue of maximal tails in \cite{BHRS}.
\begin{proposition}\label{Xprime}
An $\omega$-invariant set $X$ of ${\mathcal G}amma$ is prime if and only if for any
$\gamma_1,\gamma_2\in X$ and
any neighborhoods $U_1$, $U_2$ of $\gamma_1,\gamma_2$ respectively,
there exist $\gamma\in X$ and $\mu_1,\mu_2\in{\mathcal W}_\infty$ with
$\gamma+\omega_{\mu_1}\in U_1$ and $\gamma+\omega_{\mu_2}\in U_2$.
\varepsilonnd{proposition}
\begin{proposition}f
Suppose $X$ is a prime $\omega$-invariant set.
Take $\gamma_1,\gamma_2\in X$ and
neighborhoods $U_1$, $U_2$ of $\gamma_1,\gamma_2$ respectively.
Set $X_j={\mathcal G}amma\setminus\bigcup_{\mu\in{\mathcal W}_\infty}(U_j-\omega_\mu)$ for $j=1,2$.
Then $X_1$ and $X_2$ are $\omega$-invariant sets satisfying
$X_1\not\supset X$ and $X_2\not\supset X$.
Since $X$ is prime, we have $X_1\cup X_2\not\supset X$.
Hence there exists $\gamma\in X$ with $\gamma\notin X_1\cup X_2$.
By the definition of $X_1$ and $X_2$, there exist $\mu_1,\mu_2$ such that
$\gamma+\omega_{\mu_1}\in U_1$ and $\gamma+\omega_{\mu_2}\in U_2$.
Conversely assume that for any $\gamma_1,\gamma_2\in X$ and
any neighborhoods $U_1$, $U_2$ of $\gamma_1,\gamma_2$ respectively,
there exist $\gamma\in X$ and $\mu_1,\mu_2\in{\mathcal W}_\infty$ with
$\gamma+\omega_{\mu_1}\in U_1$ and $\gamma+\omega_{\mu_2}\in U_2$.
Take $\omega$-invariant sets $X_1$ and $X_2$ satisfying
$X_1\not\supset X$ and $X_2\not\supset X$.
There exist $\gamma_1,\gamma_2\in X$
with $\gamma_1\notin X_1$ and $\gamma_2\notin X_2$.
Hence there exist $\gamma\in X$ and $\mu_1,\mu_2\in{\mathcal W}_\infty$ with
$\gamma+\omega_{\mu_1}\notin X_1$ and $\gamma+\omega_{\mu_2}\notin X_2$.
Since $X_1$ and $X_2$ are $\omega$-invariant, we have $\gamma\notin X_1$
and $\gamma\notin X_2$.
Therefore, $X_1\cup X_2\not\supset X$.
Thus, $X$ is prime.
\varepsilonnd{proposition}f
\begin{lemma}\label{primepair0}
If an $\omega$-invariant pair $\widetilde{X}=(X,X^{(\infty)})$ is prime,
then $X^{(\infty)}=H_X$
or $X^{(\infty)}=H_X\cup\{\gamma\}$ for some $\gamma\notin H_X$.
\varepsilonnd{lemma}
\begin{proposition}f
Let $\widetilde{X}=(X,X^{(\infty)})$ be a prime $\omega$-invariant pair.
To derive a contradiction, assume $X^{(\infty)}\setminus H_X$ has
two points $\gamma_1,\gamma_2$.
Take open sets $U_1\ni\gamma_1$, $U_2\ni\gamma_2$ with $U_1\cap U_2=\varepsilonmptyset$,
$U_1\cap H_X=\varepsilonmptyset$ and $U_2\cap H_X=\varepsilonmptyset$.
Then $\widetilde{X}_i=(X,X^{(\infty)}\setminus U_i)\ (i=1,2)$
are $\omega$-invariant pairs satisfying
$\widetilde{X}=\widetilde{X}_1\cup \widetilde{X}_2$.
However, we have $\widetilde{X}\not\subset \widetilde{X}_1$ and
$\widetilde{X}\not\subset \widetilde{X}_2$.
This contradicts the primeness of $\widetilde{X}$.
\varepsilonnd{proposition}f
\begin{lemma}\label{primepair1}
An $\omega$-invariant pair $(X,H_X)$ is prime if and only if
$X$ is a prime $\omega$-invariant set.
\varepsilonnd{lemma}
\begin{proposition}f
Suppose that $(X,H_X)$ is a prime $\omega$-invariant pair.
Take $\omega$-invariant sets $X_1,X_2$ with $X\subset X_1\cup X_2$.
We have $(X,H_X)\subset (X_1,X_1)\cup (X_2,X_2)$.
Since $(X,H_X)$ is prime,
either $(X,H_X)\subset (X_1,X_1)$ or $(X,H_X)\subset (X_2,X_2)$ holds.
Therefore $X$ is a prime $\omega$-invariant set.
Conversely assume that $X$ is a prime $\omega$-invariant set.
Take two $\omega$-invariant pairs $(X_1,X_1^{(\infty)})$,
$(X_2,X_2^{(\infty)})$ with
$(X_1,X_1^{(\infty)})\cup (X_2,X_2^{(\infty)})=(X,H_X)$.
Since $X$ is prime, either $X\subset X_1$ or $X\subset X_2$.
We may assume $X\subset X_1$.
Then $X=X_1$.
Hence $H_X=H_{X_1}\subset X_1^{(\infty)}\subset H_X$.
We get $(X_1,X_1^{(\infty)})=(X,H_X)$.
By Proposition \ref{primepair}, $(X,H_X)$ is a prime $\omega$-invariant pair.
\varepsilonnd{proposition}f
\begin{lemma}\label{primepair2}
An $\omega$-invariant pair $(X,H_X\cup\{\gamma\})$ is prime
for some $\gamma\notin H_X$ if and only if $X=\gamma+\overline{\rm{sg}}(\omega)$.
\varepsilonnd{lemma}
| 3,707 | 50,987 |
en
|
train
|
0.36.5
|
\begin{proposition}f
Suppose $X$ is a prime $\omega$-invariant set.
Take $\gamma_1,\gamma_2\in X$ and
neighborhoods $U_1$, $U_2$ of $\gamma_1,\gamma_2$ respectively.
Set $X_j={\mathcal G}amma\setminus\bigcup_{\mu\in{\mathcal W}_\infty}(U_j-\omega_\mu)$ for $j=1,2$.
Then $X_1$ and $X_2$ are $\omega$-invariant sets satisfying
$X_1\not\supset X$ and $X_2\not\supset X$.
Since $X$ is prime, we have $X_1\cup X_2\not\supset X$.
Hence there exists $\gamma\in X$ with $\gamma\notin X_1\cup X_2$.
By the definition of $X_1$ and $X_2$, there exist $\mu_1,\mu_2$ such that
$\gamma+\omega_{\mu_1}\in U_1$ and $\gamma+\omega_{\mu_2}\in U_2$.
Conversely assume that for any $\gamma_1,\gamma_2\in X$ and
any neighborhoods $U_1$, $U_2$ of $\gamma_1,\gamma_2$ respectively,
there exist $\gamma\in X$ and $\mu_1,\mu_2\in{\mathcal W}_\infty$ with
$\gamma+\omega_{\mu_1}\in U_1$ and $\gamma+\omega_{\mu_2}\in U_2$.
Take $\omega$-invariant sets $X_1$ and $X_2$ satisfying
$X_1\not\supset X$ and $X_2\not\supset X$.
There exist $\gamma_1,\gamma_2\in X$
with $\gamma_1\notin X_1$ and $\gamma_2\notin X_2$.
Hence there exist $\gamma\in X$ and $\mu_1,\mu_2\in{\mathcal W}_\infty$ with
$\gamma+\omega_{\mu_1}\notin X_1$ and $\gamma+\omega_{\mu_2}\notin X_2$.
Since $X_1$ and $X_2$ are $\omega$-invariant, we have $\gamma\notin X_1$
and $\gamma\notin X_2$.
Therefore, $X_1\cup X_2\not\supset X$.
Thus, $X$ is prime.
\varepsilonnd{proposition}f
\begin{lemma}\label{primepair0}
If an $\omega$-invariant pair $\widetilde{X}=(X,X^{(\infty)})$ is prime,
then $X^{(\infty)}=H_X$
or $X^{(\infty)}=H_X\cup\{\gamma\}$ for some $\gamma\notin H_X$.
\varepsilonnd{lemma}
\begin{proposition}f
Let $\widetilde{X}=(X,X^{(\infty)})$ be a prime $\omega$-invariant pair.
To derive a contradiction, assume $X^{(\infty)}\setminus H_X$ has
two points $\gamma_1,\gamma_2$.
Take open sets $U_1\ni\gamma_1$, $U_2\ni\gamma_2$ with $U_1\cap U_2=\varepsilonmptyset$,
$U_1\cap H_X=\varepsilonmptyset$ and $U_2\cap H_X=\varepsilonmptyset$.
Then $\widetilde{X}_i=(X,X^{(\infty)}\setminus U_i)\ (i=1,2)$
are $\omega$-invariant pairs satisfying
$\widetilde{X}=\widetilde{X}_1\cup \widetilde{X}_2$.
However, we have $\widetilde{X}\not\subset \widetilde{X}_1$ and
$\widetilde{X}\not\subset \widetilde{X}_2$.
This contradicts the primeness of $\widetilde{X}$.
\varepsilonnd{proposition}f
\begin{lemma}\label{primepair1}
An $\omega$-invariant pair $(X,H_X)$ is prime if and only if
$X$ is a prime $\omega$-invariant set.
\varepsilonnd{lemma}
\begin{proposition}f
Suppose that $(X,H_X)$ is a prime $\omega$-invariant pair.
Take $\omega$-invariant sets $X_1,X_2$ with $X\subset X_1\cup X_2$.
We have $(X,H_X)\subset (X_1,X_1)\cup (X_2,X_2)$.
Since $(X,H_X)$ is prime,
either $(X,H_X)\subset (X_1,X_1)$ or $(X,H_X)\subset (X_2,X_2)$ holds.
Therefore $X$ is a prime $\omega$-invariant set.
Conversely assume that $X$ is a prime $\omega$-invariant set.
Take two $\omega$-invariant pairs $(X_1,X_1^{(\infty)})$,
$(X_2,X_2^{(\infty)})$ with
$(X_1,X_1^{(\infty)})\cup (X_2,X_2^{(\infty)})=(X,H_X)$.
Since $X$ is prime, either $X\subset X_1$ or $X\subset X_2$.
We may assume $X\subset X_1$.
Then $X=X_1$.
Hence $H_X=H_{X_1}\subset X_1^{(\infty)}\subset H_X$.
We get $(X_1,X_1^{(\infty)})=(X,H_X)$.
By Proposition \ref{primepair}, $(X,H_X)$ is a prime $\omega$-invariant pair.
\varepsilonnd{proposition}f
\begin{lemma}\label{primepair2}
An $\omega$-invariant pair $(X,H_X\cup\{\gamma\})$ is prime
for some $\gamma\notin H_X$ if and only if $X=\gamma+\overline{\rm{sg}}(\omega)$.
\varepsilonnd{lemma}
\begin{proposition}f
Suppose that an $\omega$-invariant pair $(X,H_X\cup\{\gamma\})$ is prime.
Then $(X,H_X\cup\{\gamma\})\subset (X,H_X)\cup (\gamma+\overline{\rm{sg}}(\omega),\gamma+\overline{\rm{sg}}(\omega))$
implies $(X,H_X\cup\{\gamma\})\subset(\gamma+\overline{\rm{sg}}(\omega),\gamma+\overline{\rm{sg}}(\omega))$
because $H_X\cup\{\gamma\}\not\subset H_X$.
Hence $\gamma+\overline{\rm{sg}}(\omega)\subset X\subset\gamma+\overline{\rm{sg}}(\omega)$.
Thus, we get $X=\gamma+\overline{\rm{sg}}(\omega)$.
Conversely, assume $X=\gamma+\overline{\rm{sg}}(\omega)$.
Take two $\omega$-invariant pairs $(X_1,X_1^{(\infty)})$,
$(X_2,X_2^{(\infty)})$ with
$(X_1,X_1^{(\infty)})\cup (X_2,X_2^{(\infty)})=(X,H_X\cup\{\gamma\})$.
We may assume $\gamma\in X_1^{(\infty)}$.
Then we have $X=\gamma+\overline{\rm{sg}}(\omega)\subset X_1^{(\infty)}+\overline{\rm{sg}}(\omega)\subset X_1\subset X$.
Hence $X_1=X$.
We have $H_X\cup\{\gamma\}=H_{X_1}\cup\{\gamma\}\subset
X_1^{(\infty)}\subset H_X\cup\{\gamma\}$.
Therefore $(X_1,X_1^{(\infty)})=(X,H_X\cup\{\gamma\})$.
By Proposition \ref{primepair}, $(X,H_X\cup\{\gamma\})$ is a prime $\omega$-invariant pair.
\varepsilonnd{proposition}f
\begin{proposition}\label{pp}
An $\omega$-invariant pair $(X,X^{(\infty)})$ is prime if and only if
either $X$ is prime and $X^{(\infty)}=H_X$ or $X=\gamma+\overline{\rm{sg}}(\omega)$ and
$X^{(\infty)}=H_X\cup\{\gamma\}$ for some $\gamma\notin H_X$.
\varepsilonnd{proposition}
\begin{proposition}f
Combine Lemma \ref{primepair0}, Lemma \ref{primepair1}
and Lemma \ref{primepair2}.
\varepsilonnd{proposition}f
| 2,071 | 50,987 |
en
|
train
|
0.36.6
|
\section{The ideal structure of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ (part 1)}
In this section and the next section,
we completely determine the ideal structure of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$
(Theorem \ref{idestr1}, Theorem \ref{idestr2}).
The ideal structure of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ depends on whether $\omega\in{\mathcal G}amma^\infty$ satisfies
the following condition:
\begin{corollary}n\label{cond}
For each $i\in\mathbb{Z}_+$, one of the following two conditions is satisfied:
\benu
\item For any positive integer $k$, $k\omega_i\neq 0$.
\item There exists a sequence $\mu_1,\mu_2,\ldots$ in ${\mathcal W}_\infty$ such that
$S_{\mu_k}^*S_i=0$ for any $k$ and $\lim_{k\to\infty}\omega_{\mu_k}=0$.
\varepsilonnd{enumerate}
\varepsilonnd{corollary}n
This condition is an analogue of Condition (K) in the case of
graph algebras \cite{BHRS}.
In this section,
we deal with the case that $\omega$ satisfies Condition \ref{cond}.
\begin{proposition}\label{cond.exp2}
If $\omega$ satisfies Condition \ref{cond},
then for an ideal $I$ that is not ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$,
there exists a unique conditional expectation $E_I$ from $({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I$ onto
${\mathcal F}_I$ such that
$E_I(S_\mu fS_\nu^*)=\delta_{|\mu|,|\nu|}S_\mu fS_\nu^*$
for $\mu,\nu\in{\mathcal W}_\infty,\ f\in C_0(X_I)$.
\varepsilonnd{proposition}
\begin{proposition}f
Take $x=\sum_{l=1}^LS_{\mu_l}f_lS_{\nu_l}^*\in ({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I$ where
$\mu_l,\nu_l\in{\mathcal W}_\infty$ and $f_l\in C_0(X_I)$ for $l=1,2,\ldots,L$.
Set $x_0=\sum_{|\mu_l|=|\nu_l|}S_{\mu_l}f_lS_{\nu_l}^*$
and we will prove that $\|x_0\|\leq\|x\|$.
If we choose a positive integer $n$ so that $|\mu_l|,|\nu_l|\leq n$ and
$\mu_l,\nu_l\in{\mathcal W}_n$ for $l=1,2,\ldots,L$,
then $x_0\in{\mathcal F}_I^{(n)}$.
By Lemma \ref{F_n},
there exist $x_0^{(k)}\in{\mathcal F}_{I}^{(n,k)}\ (0\leq k\leq n-1)$
and $x_0^{(n)}\in{\mathcal G}_{I}^{(n,n)}$ such that $x_0=\sum_{k=0}^nx_0^{(k)}$.
We have $\|x_0\|=\max\{\|x_0^{(0)}\|,\ldots,\|x_0^{(n)}\|\}$.
First we consider the case that $\|x_0\|=\|x_0^{(k)}\|$
for some $k\leq n-1$.
If we set $q_k=\sum_{\mu\in{\mathcal W}_n^{(k)}}S_\mu p_nS_\mu^*\in M(({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I)$,
then $q_k$ is a projection satisfying that
$q_kS_{\mu_l}S_{\nu_l}^*q_k=0$ if $|\mu_l|\neq |\nu_l|$.
Hence $q_kxq_k=q_kx_0q_k=x_0^{(k)}$.
We get $\|x_0\|=\|x_0^{(k)}\|=\|q_kxq_k\|\leq\|x\|$.
Next we consider the case that $\|x_0\|=\|x_0^{(n)}\|$.
Then there exists $\gamma_0\in X_I$
such that $\|x_0^{(n)}\|=\|x_0^{(n)}(\gamma_0)\|$.
By Lemma \ref{X^{(n)}} (iv), we have
$$X_I=\bigcup_{\mu\in{\mathcal W}_n}(X_I^{(n)}+\omega_\mu)\cup
\bigcap_{k=1}^\infty \bigg( \bigcup_{\mu\in{\mathcal W}_n^{(k)}}(X_I+\omega_\mu)\bigg).$$
When $\gamma_0\in X_I^{(n)}+\omega_\mu$ for some $\mu\in{\mathcal W}_n$,
set $u=\sum_{\nu\in{\mathcal W}_n^{(n)}}S_\nu S_\mu p_nS_\nu^*\in M(({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I)$.
Then $u$ is a partial isometry.
We have $u^*xu=u^*x_0u=u^*x_0^{(n)}u=\pi_n(\sigma_{\omega_\mu}(x_0^{(n)}))$
where $\pi_n$ is the natural surjection
from ${\mathcal G}_I^{(n,n)}$ onto ${\mathcal F}_I^{(n,n)}$.
Since $\gamma_0-\omega_\mu\in X_I^{(n)}$, we have
$$\|\pi_n(\sigma_{\omega_\mu}(x_0^{(n)}))\|
\geq\|\sigma_{\omega_\mu}(x_0^{(n)})(\gamma_0-\omega_\mu)\|
=\|x_0^{(n)}(\gamma_0)\|=\|x_0^{(n)}\|=\|x_0\|.$$
Therefore $\|x_0\|\leq\|u^*xu\|\leq\|x\|$.
When $\gamma_0\in\bigcap_{k=1}^\infty
\big( \bigcup_{\mu\in{\mathcal W}_n^{(k)}}(X_I+\omega_\mu)\big)$,
we can find $i\in\{1,2,\ldots,n\}$ such that
$\gamma_0-k\omega_i\in X_I$ for all $k\in\mathbb{N}$.
Since $\omega$ satisfies Condition \ref{cond},
either $k\omega_i\neq 0$ for any $k\in\mathbb{Z}_+$ or there exists a sequence
$\{\mu_k\}_{k\in\mathbb{Z}_+}\subset{\mathcal W}_n$ with $\lim_{k\to\infty}\omega_{\mu_k}=0$
and $S_{\mu_k}^*S_i=0$ for any $k$.
In the case that $k\omega_i\neq 0$ for any $k\in\mathbb{Z}_+$,
we can find a neighborhood $U$ of $\gamma_0-n\omega_i\in X_I$ such that
$U\cap (U+k\omega_i)=\varepsilonmptyset$ for $k=1,2,\ldots,n$.
Choose a function $f$ with $0\leq f\leq 1$ satisfying that
the support of $f$ is contained in $U$ and $f(\gamma_0-n\omega_i)=1$.
Set $u=\sum_{\mu\in{\mathcal W}_n^{(n)}}S_\mu S_i^n f^{1/2} S_\mu^*\in({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I$.
Since
$$u^*u=\sum_{\mu,\nu\in{\mathcal W}_n^{(n)}}S_\mu f^{1/2} {S_i^*}^n S_\mu^*
S_\nu S_i^n f^{1/2}S_\nu^*=\sum_{\mu\in{\mathcal W}_n^{(n)}}S_\mu f S_\mu^*,$$
$u^*u$ corresponds to $1\otimes f$
under the isomorphism ${\mathcal G}_I^{(n,n)}\cong\mathbb{M}_{n^n}\otimes C_0(X_I)$.
Thus we have $\|u^*u\|=\sup_{\gamma\in X_I}|f(\gamma)|=1$,
and so $\|u\|=1$.
A routine computation shows that
$u^*xu=u^*x_0^{(n)}u=f\sigma_{n\omega_i}x_0^{(n)}\in C_0(X_I,\mathbb{M}_{n^n})$.
Since $\gamma_0-n\omega_i\in X_I$, we have
$$\|u^*xu\|\geq\|f(\gamma_0-n\omega_i)\sigma_{n\omega_i}x_0^{(n)}(\gamma_0-n\omega_i)\|=\|x_0^{(n)}(\gamma_0)\|=\|x_0\|.$$
Hence $\|x_0\|\leq\|u^*xu\|\leq\|x\|$.
Finally, we consider the case that there exists a sequence
$\{\mu_k\}_{k\in\mathbb{Z}_+}\subset{\mathcal W}_n$
with $\lim_{k\to\infty}\omega_{\mu_k}=0$
and $S_{\mu_k}^*S_i=0$ for any $k\in\mathbb{Z}_+$.
For $k\in\mathbb{Z}_+$, define a partial isometry $u_k=\sum_{\mu\in{\mathcal W}_n^{(n)}}S_\mu S_i^nS_{\mu_k}S_\mu^*\in{\mathcal O}_\infty\subset M(({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I)$.
A routine computation shows that
$u_k^*xu_k=u_k^*x_0^{(n)}u_k
=\sigma_{n\omega_i+\omega_{\mu_k}}x_0^{(n)}\in C_0(X_I,M_{n^n})$.
Since $\gamma_0-n\omega_i\in X_I$, we have
$$\|u_k^*xu_k\|\geq\|\sigma_{n\omega_i+\omega_{\mu_k}}x_0^{(n)}(\gamma_0-n\omega_i)\|=\|x_0^{(n)}(\gamma_0+\omega_{\mu_k})\|.$$
Hence we have
$\|x_0^{(n)}(\gamma_0+\omega_{\mu_k})\|\leq\|u_k^*xu_k\|\leq\|x\|$
for any $k\in\mathbb{Z}_+$.
Therefore $\|x_0\|=\|x_0^{(n)}(\gamma_0)\|=\lim_{k\to\infty}\|x_0^{(n)}(\gamma_0+\omega_{\mu_k})\|\leq\|x\|$.
Hence the map
\begin{align*}
\spa\{S_\mu fS_\nu^*&\mid\mu,\nu\in{\mathcal W}_\infty,\ f\in C_0(X_I)\}\ni x\\
&\mapsto\ x_0\in \spa\{S_\mu fS_\nu^*\mid\mu,\nu\in{\mathcal W}_\infty,\ |\mu|=|\nu|,\ f\in C_0(X_I)\}.
\varepsilonnd{align*}
is well-defined and norm-decreasing.
The extension $E_I$ of the map above is the desired conditional expectation
onto ${\mathcal F}_I$.
Uniqueness is easy to verify.
\varepsilonnd{proposition}f
By uniqueness, the conditional expectation $E_{I}$ above coincides
with the one in Lemma \ref{cond.exp1} when $I$ is gauge invariant.
Actually an ideal of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ is gauge invariant
if there exists such a conditional expectation,
as we see in the proof of the following theorem.
\begin{theorem}\label{idestr1}
Suppose that $\omega$ satisfies Condition \ref{cond}.
Then for any ideal $I$ we have $I_{\widetilde{X}_I}=I$,
and so $I$ is gauge invariant.
Hence there is a one-to-one correspondence between the set of ideals of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$
and the set of $\omega$-invariant pairs of subsets of ${\mathcal G}amma$.
\varepsilonnd{theorem}
\begin{proposition}f
If $X_I=\varepsilonmptyset$, then $I={\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ so $I_{\widetilde{X}_I}=I$.
Let $I$ be an ideal that is not ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$, and set $J=I_{\widetilde{X}_I}$.
By the same way as in the proof of Proposition \ref{unique},
we can find a surjective $*$-homomorphism $\pi:({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/J\to({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I$
whose restriction on ${\mathcal F}_{J}$ is an isomorphism from
${\mathcal F}_{J}$ onto ${\mathcal F}_{I}$.
By Proposition \ref{cond.exp2},
there exists a conditional expectation $E_I:({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I\to{\mathcal F}_{I}$
satisfying $E_I\circ \pi=\pi\circ E_{J}$,
where $E_J:({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I\to{\mathcal F}_{J}$ is a faithful conditional expectation
defined in Lemma \ref{cond.exp1}.
By Lemma \ref{isom}, $\pi$ is injective.
Therefore $I=I_{\widetilde{X}_I}$.
The last part follows from Theorem \ref{OneToOne}.
\varepsilonnd{proposition}f
\begin{corollary}\label{primitive1}
When $\omega$ satisfies Condition \ref{cond},
an ideal $I$ of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ is primitive
if and only if the $\omega$-invariant pair $\widetilde{X}_I$ is prime.
\varepsilonnd{corollary}
\begin{proposition}f
It follows from Proposition \ref{primepair} and Theorem \ref{idestr1}.
\varepsilonnd{proposition}f
| 3,548 | 50,987 |
en
|
train
|
0.36.7
|
\section{The ideal structure of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ (part 2)}
In this section, we investigate the ideal structure of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$
when $\omega$ does not satisfy Condition \ref{cond}
i.e.\ there exists $i\in\mathbb{Z}_+$ such that
$k\omega_i=0$ for some positive integer $k$,
and that there exist no sequences $\mu_1,\mu_2,\ldots$ in ${\mathcal W}_\infty$ such that
$S_{\mu_k}^*S_i=0$ for any $k$ and $\lim_{k\to\infty}\omega_{\mu_k}=0$.
Note that such $i$ is unique.
Without loss of generality, we may assume $i=1$.
Let $K$ be the smallest positive integer satisfying $K\omega_1=0$.
Denote by ${\mathcal G}amma'$ the quotient of ${\mathcal G}amma$
by the subgroup generated by $\omega_1$, which is isomorphic to $\mathbb{Z}/K\mathbb{Z}$.
We denote by $[\gamma]$ and $[U]$ the images in ${\mathcal G}amma'$
of $\gamma\in{\mathcal G}amma$ and $U\subset{\mathcal G}amma$ respectively.
We use the symbol $([\gamma],\theta)$
for denoting elements of ${\mathcal G}amma'\times\mathbb{T}$.
Define $A=\cspa\{S_1^kf{S_1^*}^l\mid f\in C_0({\mathcal G}amma), k,l\in\mathbb{N}\}$
which is a $C^*$-subalgebra of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$.
In \cite{Ka1},
we defined a $C^*$-algebra $T_K$ and a continuous family of $*$-homomorphisms
$\varphi_\gamma:A\to T_K$ for $\gamma\in{\mathcal G}amma$.
Note that $\varphi_\gamma(x)=0$
if and only if $\varphi_{\gamma+\omega_1}(x)=0$ for $x\in A$.
We also defined $\psi_{\gamma,\theta}=\pi_\theta\circ \varphi_\gamma$
for $(\gamma,\theta)\in{\mathcal G}amma\times\mathbb{T}$,
where $\pi_\theta:T_K\to\mathbb{M}_{K}$ is a continuous family of $*$-homomorphisms.
\begin{definition}\rm
For an ideal $I$ of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$,
we define the closed subset $Y_I$ of ${\mathcal G}amma'\times\mathbb{T}$ by
$$Y_I=\{([\gamma],\theta)\in{\mathcal G}amma'\times\mathbb{T}\mid \psi_{\gamma,\theta}(x)=0
\mbox{ for all }x\in A\cap I\}.$$
We denote by $\widetilde{Y}_I$ the pair $(Y_I,X_I^{(\infty)})$ of
a subset $Y_I$ of ${\mathcal G}amma'\times\mathbb{T}$
and a subset $X_I^{(\infty)}$ of ${\mathcal G}amma$.
\varepsilonnd{definition}
\begin{definition}\rm\label{omegaY}
For a pair $\widetilde{Y}=(Y,X^{(\infty)})$ of
a subset $Y$ of ${\mathcal G}amma'\times\mathbb{T}$
and a subset $X^{(\infty)}$ of ${\mathcal G}amma$,
we define subsets $X$ and $X^{(n)}$ of ${\mathcal G}amma$ by
\begin{align*}
X&=\{\gamma\in{\mathcal G}amma\mid ([\gamma],\theta)\in Y
\mbox{ for some } \theta\in\mathbb{T}\},\\
X^{(n)}&=X^{(\infty)}\cup\bigcup_{i=n+1}^\infty(X+\omega_i).
\varepsilonnd{align*}
With this notation,
a pair $\widetilde{Y}=(Y,X^{(\infty)})$ is called {\varepsilonm $\omega$-invariant}
if $(X,X^{(\infty)})$ is an $\omega$-invariant pair of subsets of ${\mathcal G}amma$
and if $Y$ is a closed set satisfying that $[X^{(1)}]\times\mathbb{T}\subset Y$.
\varepsilonnd{definition}
\begin{proposition}
For an ideal $I$ of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$,
the pair $\widetilde{Y}_I$ is $\omega$-invariant.
\varepsilonnd{proposition}
\begin{proposition}f
By \cite[Proposition 5.15]{Ka1}, we have
$$X_I=\{\gamma\in{\mathcal G}amma\mid
([\gamma],\theta)\in Y_I\mbox{ for some } \theta\in\mathbb{T}\}.$$
By the argument in the proof of \cite[Lemma 5.21]{Ka1}, we have
$$X_I^{(1)}=\{\gamma\in{\mathcal G}amma\mid
\varphi_{\gamma}(x)=0\mbox{ for any } x\in A\cap I\}.$$
Therefore $[X_I^{(1)}]\times\mathbb{T}\subset Y_I$.
Thus the pair $\widetilde{Y}_I$ is $\omega$-invariant.
\varepsilonnd{proposition}f
We get the $\omega$-invariant pair $\widetilde{Y}_I$
from an ideal $I$ of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$.
Conversely, from an $\omega$-invariant pair $\widetilde{Y}$ ,
we can construct the ideal $I_{\widetilde{Y}}$ of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$.
\begin{definition}\rm
For an $\omega$-invariant pair $\widetilde{Y}=(Y,X^{(\infty)})$,
we define $J_{\widetilde{Y}}\subset A$
and $I_{\widetilde{Y}}\subset{\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ by
\begin{align*}
J_{\widetilde{Y}}&=\{x\in A\mid
\psi_{\gamma,\theta}(x)=0\mbox{ for }([\gamma],\theta)\in Y,
\mbox{ and } \varphi_\gamma(x)=0\mbox{ for }\gamma\in X^{(1)}\},\\
I_{\widetilde{Y}}
&=\cspa\big(\{S_\mu xS_\nu^*\mid \mu,\nu\in{\mathcal W}_\infty,\
x\in J_{\widetilde{Y}}\}\\
&\hspace*{2cm}\cup\{S_\mu p_nfS_\nu \mid \mu,\nu\in{\mathcal W}_\infty,\
n\in\mathbb{Z}_+,\ f\in C_0({\mathcal G}amma\setminus X^{(n)})\}\big),
\varepsilonnd{align*}
with the notation in Definition \ref{omegaY}.
\varepsilonnd{definition}
\begin{proposition}
For an $\omega$-invariant pair $\widetilde{Y}$,
$I_{\widetilde{Y}}$ is an ideal of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$.
\varepsilonnd{proposition}
\begin{proposition}f
Once noting that
$J_{\widetilde{Y}}\cap C_0({\mathcal G}amma)=C_0({\mathcal G}amma\setminus X)$ and
$J_{\widetilde{Y}}\cap p_1 C_0({\mathcal G}amma)=p_1 C_0({\mathcal G}amma\setminus X^{(1)})$,
we can prove that $I_{\widetilde{Y}}$ is an ideal
in a similar way to Proposition \ref{I_X,Xinfty}
with the help of the computation in \cite[Proposition 5.20]{Ka1}.
\varepsilonnd{proposition}f
\begin{lemma}\label{Y_I_Y}
Let $\widetilde{Y}=(Y,X^{(\infty)})$ be an $\omega$-invariant pair.
For any $([\gamma],\theta)\not\in Y$, there exists
$x\in J_{\widetilde{Y}}$ such that $\psi_{\gamma,\theta}(x)\neq 0$.
\varepsilonnd{lemma}
\begin{proposition}f
The proof goes exactly the same as in the proof of \cite[Lemma 5.22]{Ka1},
once noting that $([\gamma],\theta)\not\in Y$ implies $\gamma\not\in X^{(1)}$.
\varepsilonnd{proposition}f
\begin{proposition}\label{exist2}
Let $\widetilde{Y}=(Y,X^{(\infty)})$ be an $\omega$-invariant pair,
and set $I=I_{\widetilde{Y}}$.
Then we have $\widetilde{Y}_I=\widetilde{Y}$.
\varepsilonnd{proposition}
\begin{proposition}f
By Lemma \ref{Y_I_Y}, we get $Y_I\subset Y$.
To prove the other inclusion, it is sufficient to see
that $\psi_{\gamma,\theta}(x)=0$
for $([\gamma],\theta)\in Y$ and $x\in I\cap A$.
Take $\varepsilon>0$ arbitrarily.
Since $x\in I$,
there exist $\mu_l,\nu_l\in{\mathcal W}_\infty$, $x_l\in J_{\widetilde{Y}}$
for $l=1,2,\ldots,L$ and $\mu_k',\nu_k'\in{\mathcal W}_\infty$, $n_k\in\mathbb{Z}_+$,
$f_k\in C_0({\mathcal G}amma\setminus X^{(n_k)})$ for $k=1,2,\ldots,K$
such that
$$\bigg\|x-\sum_{l=1}^LS_{\mu_l}x_lS_{\nu_l}^*-\sum_{k=1}^K
S_{\mu_k'}p_{n_k}f_kS_{\nu_k'}^*\bigg\|<\varepsilon.$$
Take a positive integer $m$ such that $m\geq |\mu_l|,|\nu_l|$
for any $l$ and $m> |\mu_k'|,|\nu_k'|$ for any $k$.
Then, $\left\|{S_1^*}^mxS_1^m-\sum_{l=1}^Lx_l'\right\|<\varepsilon$
where $x_l'={S_1^*}^mS_{\mu_l}x_lS_{\nu_l}^*S_1^m$ for $l=1,2,\ldots,L$.
Since $x_l'\in J_{\widetilde{Y}}$,
we have $\|\psi_{\gamma,\theta}({S_1^*}^mxS_1^m)\|<\varepsilon$.
Since $\psi_{\gamma,\theta}(S_1)$ is a unitary,
we have $\|\psi_{\gamma,\theta}(x)\|<\varepsilon$ for arbitrary $\varepsilon>0$.
Hence, we have $\psi_{\gamma,\theta}(x)=0$.
Therefore we get $Y_I=Y$.
From $Y_I=Y$, we have $X_I=X$.
By the definition of $I$,
we see that $X_I^{(n)}\subset X^{(n)}$ for $n\in\mathbb{Z}_+$.
To the contrary, assume that $X_I^{(n)}\subsetneqq X^{(n)}$.
Then there exists $f\in C_0({\mathcal G}amma)$
such that $p_nf\in I$ and $f(\gamma_0)=1$ for some $\gamma_0\in X^{(n)}$.
Since $p_nf\in I$, there exist
$\mu_l,\nu_l\in{\mathcal W}_\infty$, $x_l\in J_{\widetilde{Y}}$
for $l=1,2,\ldots,L$ and $\mu_k',\nu_k'\in{\mathcal W}_\infty$, $n_k\in\mathbb{Z}_+$,
$f_k\in C_0({\mathcal G}amma\setminus X^{(n_k)})$ for $k=1,2,\ldots,K$
such that
$$\bigg\|p_nf-\sum_{l=1}^LS_{\mu_l}x_lS_{\nu_l}^*-\sum_{k=1}^K
S_{\mu_k'}p_{n_k}f_kS_{\nu_k'}^*\bigg\|<\frac12.$$
Take a positive integer $m$ so large that
$\mu_l,\nu_l,\mu_k',\nu_k'\in{\mathcal W}_{m}$, $n_k\leq m$
for any $l,k$ and $n\leq m$.
By Lemma \ref{X^{(n)}} (iii), we have
$X^{(n)}=X^{(m)}\cup\bigcup_{i=n+1}^{m}(X+\omega_i)$.
We first consider the case that $\gamma_0\in X^{(m)}$.
By \cite[Lemma 5.4]{Ka1}, there exists
$g_l\in C_0({\mathcal G}amma\setminus X^{1})$ with $p_1x_lp_1=p_1g_l$ for any $l$.
Hence we have $p_{m}x_lp_{m}=p_{m}p_1x_lp_1p_{m}=p_{m}g_l$ for any $l$.
Since
$$p_{m}\bigg(p_nf-\sum_{l=1}^LS_{\mu_l}x_lS_{\nu_l}^*
-\sum_{k=1}^K S_{\mu_k'}p_{n_k}f_kS_{\nu_k'}^*\bigg)p_{m}=
p_{m}f-\sum_{\mu_l=\nu_l=\varepsilonmptyset}p_{m}g_l
-\sum_{\mu_k'=\nu_k'=\varepsilonmptyset}p_{m}f_k,$$
we get
$\|f-\sum_{\mu_l=\nu_l=\varepsilonmptyset}g_l-\sum_{\mu_k'=\nu_k'=\varepsilonmptyset}f_k\|<1/2$.
This contradicts the fact that $f(\gamma_0)=1$, $g_l(\gamma_0)=0$ and
$f_k(\gamma_0)=0$ for any $l,k$.
When $\gamma_0\in X+\omega_i$ for some $i$ with $n<i\leq m$,
we have $\sigma_{\omega_i}f=S_i^*(p_nf)S_i\in I$ and
$\sigma_{\omega_i}f(\gamma_0-\omega_i)=1$.
This contradicts the fact that $X_I=X$.
Therefore $X_I^{(n)}=X^{(n)}$ for a positive integer $n$.
Hence $X_I^{(\infty)}=\bigcap_{n=1}^\infty X_I^{(n)}=
\bigcap_{n=1}^\infty X^{(n)}=X^{(\infty)}$.
Thus we have $\widetilde{Y}_I=\widetilde{Y}$.
\varepsilonnd{proposition}f
\begin{corollary}\label{Ysub}
For two $\omega$-invariant pairs $\widetilde{Y}_1=(Y_1,X_1^{(\infty)})$,
$\widetilde{Y}_2=(Y_2,X_2^{(\infty)})$,
we have $I_{\widetilde{Y}_1}\subset I_{\widetilde{Y}_2}$
if and only if $Y_1\supset Y_2$ and $X_1^{(\infty)}\supset X_2^{(\infty)}$.
\varepsilonnd{corollary}
A relation between $I_{\widetilde{Y}}$ and $I_{\widetilde{X}}$
can be described as follows.
\begin{proposition}\label{rotation}
Let $\widetilde{Y}=(Y,X^{(\infty)})$ be an $\omega$-invariant pair.
For $t\in\mathbb{T}$, set $\widetilde{Y}_t=(Y_t,X^{(\infty)})$ where
$Y_t=\{([\gamma],\theta)\in{\mathcal G}amma'\times\mathbb{T}\mid ([\gamma],t\theta)\in Y\}$.
Then $\widetilde{Y}_t$ is $\omega$-invariant
and $\beta_t(I_{\widetilde{Y}})=I_{\widetilde{Y}_{t^K}}$
where $\beta$ is the gauge action.
We also have $I_{\widetilde{X}}
=\bigcap_{t\in\mathbb{T}}I_{\widetilde{Y}_t}$
where $\widetilde{X}=(X,X^{(\infty)})$ and
$X=\{\gamma\in{\mathcal G}amma\mid ([\gamma],\theta)\in Y
\mbox{ for some }\theta\in\mathbb{T}\}$.
\varepsilonnd{proposition}
| 4,069 | 50,987 |
en
|
train
|
0.36.8
|
From $Y_I=Y$, we have $X_I=X$.
By the definition of $I$,
we see that $X_I^{(n)}\subset X^{(n)}$ for $n\in\mathbb{Z}_+$.
To the contrary, assume that $X_I^{(n)}\subsetneqq X^{(n)}$.
Then there exists $f\in C_0({\mathcal G}amma)$
such that $p_nf\in I$ and $f(\gamma_0)=1$ for some $\gamma_0\in X^{(n)}$.
Since $p_nf\in I$, there exist
$\mu_l,\nu_l\in{\mathcal W}_\infty$, $x_l\in J_{\widetilde{Y}}$
for $l=1,2,\ldots,L$ and $\mu_k',\nu_k'\in{\mathcal W}_\infty$, $n_k\in\mathbb{Z}_+$,
$f_k\in C_0({\mathcal G}amma\setminus X^{(n_k)})$ for $k=1,2,\ldots,K$
such that
$$\bigg\|p_nf-\sum_{l=1}^LS_{\mu_l}x_lS_{\nu_l}^*-\sum_{k=1}^K
S_{\mu_k'}p_{n_k}f_kS_{\nu_k'}^*\bigg\|<\frac12.$$
Take a positive integer $m$ so large that
$\mu_l,\nu_l,\mu_k',\nu_k'\in{\mathcal W}_{m}$, $n_k\leq m$
for any $l,k$ and $n\leq m$.
By Lemma \ref{X^{(n)}} (iii), we have
$X^{(n)}=X^{(m)}\cup\bigcup_{i=n+1}^{m}(X+\omega_i)$.
We first consider the case that $\gamma_0\in X^{(m)}$.
By \cite[Lemma 5.4]{Ka1}, there exists
$g_l\in C_0({\mathcal G}amma\setminus X^{1})$ with $p_1x_lp_1=p_1g_l$ for any $l$.
Hence we have $p_{m}x_lp_{m}=p_{m}p_1x_lp_1p_{m}=p_{m}g_l$ for any $l$.
Since
$$p_{m}\bigg(p_nf-\sum_{l=1}^LS_{\mu_l}x_lS_{\nu_l}^*
-\sum_{k=1}^K S_{\mu_k'}p_{n_k}f_kS_{\nu_k'}^*\bigg)p_{m}=
p_{m}f-\sum_{\mu_l=\nu_l=\varepsilonmptyset}p_{m}g_l
-\sum_{\mu_k'=\nu_k'=\varepsilonmptyset}p_{m}f_k,$$
we get
$\|f-\sum_{\mu_l=\nu_l=\varepsilonmptyset}g_l-\sum_{\mu_k'=\nu_k'=\varepsilonmptyset}f_k\|<1/2$.
This contradicts the fact that $f(\gamma_0)=1$, $g_l(\gamma_0)=0$ and
$f_k(\gamma_0)=0$ for any $l,k$.
When $\gamma_0\in X+\omega_i$ for some $i$ with $n<i\leq m$,
we have $\sigma_{\omega_i}f=S_i^*(p_nf)S_i\in I$ and
$\sigma_{\omega_i}f(\gamma_0-\omega_i)=1$.
This contradicts the fact that $X_I=X$.
Therefore $X_I^{(n)}=X^{(n)}$ for a positive integer $n$.
Hence $X_I^{(\infty)}=\bigcap_{n=1}^\infty X_I^{(n)}=
\bigcap_{n=1}^\infty X^{(n)}=X^{(\infty)}$.
Thus we have $\widetilde{Y}_I=\widetilde{Y}$.
\varepsilonnd{proposition}f
\begin{corollary}\label{Ysub}
For two $\omega$-invariant pairs $\widetilde{Y}_1=(Y_1,X_1^{(\infty)})$,
$\widetilde{Y}_2=(Y_2,X_2^{(\infty)})$,
we have $I_{\widetilde{Y}_1}\subset I_{\widetilde{Y}_2}$
if and only if $Y_1\supset Y_2$ and $X_1^{(\infty)}\supset X_2^{(\infty)}$.
\varepsilonnd{corollary}
A relation between $I_{\widetilde{Y}}$ and $I_{\widetilde{X}}$
can be described as follows.
\begin{proposition}\label{rotation}
Let $\widetilde{Y}=(Y,X^{(\infty)})$ be an $\omega$-invariant pair.
For $t\in\mathbb{T}$, set $\widetilde{Y}_t=(Y_t,X^{(\infty)})$ where
$Y_t=\{([\gamma],\theta)\in{\mathcal G}amma'\times\mathbb{T}\mid ([\gamma],t\theta)\in Y\}$.
Then $\widetilde{Y}_t$ is $\omega$-invariant
and $\beta_t(I_{\widetilde{Y}})=I_{\widetilde{Y}_{t^K}}$
where $\beta$ is the gauge action.
We also have $I_{\widetilde{X}}
=\bigcap_{t\in\mathbb{T}}I_{\widetilde{Y}_t}$
where $\widetilde{X}=(X,X^{(\infty)})$ and
$X=\{\gamma\in{\mathcal G}amma\mid ([\gamma],\theta)\in Y
\mbox{ for some }\theta\in\mathbb{T}\}$.
\varepsilonnd{proposition}
\begin{proposition}f
See \cite[Proposition 5.24]{Ka1}.
\varepsilonnd{proposition}f
\begin{proposition}
For an $\omega$-invariant pair $\widetilde{X}=(X,X^{(\infty)})$
of subsets of ${\mathcal G}amma$,
the pair $\widetilde{Y}=([X]\times\mathbb{T},X^{(\infty)})$ is $\omega$-invariant
and $I_{\widetilde{Y}}=I_{\widetilde{X}}$.
\varepsilonnd{proposition}
\begin{proposition}f
Obvious by Proposition \ref{rotation}.
\varepsilonnd{proposition}f
Now, we turn to showing that $I_{\widetilde{Y}_I}=I$ for any ideal $I$
(Theorem \ref{idestr2}).
To see this, we examine the primitive ideal space of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$.
Set $\overline{\rm{sg}}_1(\omega)=\overline{\rm{sg}}(\omega)\setminus\{0,\omega_1,\ldots,(K-1)\omega_1\}$.
\begin{lemma}\label{isolate}
We have $\overline{\rm{sg}}_1(\omega)=\overline{\bigcup_{i=2}^\infty({\rm{sg}}(\omega)+\omega_i)}$ and
$\overline{\rm{sg}}_1(\omega)$ is an $\omega$-invariant set.
\varepsilonnd{lemma}
\begin{proposition}f
For $\gamma\in\overline{\rm{sg}}(\omega)$, we can find $\mu_k\in{\mathcal W}_\infty$ such that
$\gamma=\lim_{k\to\infty}\omega_{\mu_k}$.
If $\mu_k=(1,1,\ldots,1)$ for sufficiently large $k$,
then $\gamma=m\omega_1$ for some $m\in\mathbb{N}$.
Hence for $\gamma\in\overline{\rm{sg}}_1(\omega)$, we can find $\mu_k\in{\mathcal W}_\infty$
with $\omega_{\mu_k}\in\bigcup_{i=2}^\infty({\rm{sg}}(\omega)+\omega_i)$ such that
$\gamma=\lim_{k\to\infty}\omega_{\mu_k}$.
Thus $\overline{\rm{sg}}_1(\omega)\subset\overline{\bigcup_{i=2}^\infty({\rm{sg}}(\omega)+\omega_i)}$.
To prove the other inclusion,
suppose $m\omega_1\in\overline{\bigcup_{i=2}^\infty({\rm{sg}}(\omega)+\omega_i)}$
for some $0\leq m<K$
and we will derive a contradiction.
In this case, $0$ is also in $\overline{\bigcup_{i=2}^\infty({\rm{sg}}(\omega)+\omega_i)}$.
Hence there exists a sequence $\{\mu_k\}$ in ${\mathcal W}_\infty$
with $S_{\mu_k}^*S_1=0$
such that $0=\lim_{k\to\infty}\omega_{\mu_k}$.
This contradicts the fact that $\omega$ does not satisfy Condition \ref{cond}.
Therefore $\overline{\rm{sg}}_1(\omega)=\overline{\bigcup_{i=2}^\infty({\rm{sg}}(\omega)+\omega_i)}$.
From this equality, it is easy to see that
$\overline{\rm{sg}}_1(\omega)$ is an $\omega$-invariant set.
\varepsilonnd{proposition}f
\begin{corollary}\label{cptnbhd}
For any $\gamma_0\in{\mathcal G}amma$,
there exists a compact neighborhood $X$ of $\gamma_0$ satisfying that
$X\cap(X+\gamma)=\varepsilonmptyset$ for any $\gamma\in\overline{\rm{sg}}(\omega)\setminus\{0\}$.
\varepsilonnd{corollary}
\begin{proposition}f
Since
$\overline{\rm{sg}}(\omega)\setminus\{0\}=\overline{\rm{sg}}_1(\omega)\cup\{\omega_1,2\omega_1,\ldots,(K-1)\omega_1\}$
is closed by Lemma \ref{isolate}, there exists a neighborhood $U$
of $0$ with $U\cap (\overline{\rm{sg}}(\omega)\setminus\{0\})=\varepsilonmptyset$.
If we take a compact neighborhood $V$ of $0$ such that $V-V\subset U$,
then $X=\gamma_0+V$ becomes a desired compact neighborhood of $\gamma_0$.
\varepsilonnd{proposition}f
\begin{lemma}\label{H_X}
For an $\omega$-invariant set $X$, we have
$H_X=\bigcap_{n=1}^\infty\overline{\bigcup_{i=n}^\infty(X+\omega_i)}$.
If two $\omega$-invariant sets $X_1$ and $X_2$ satisfy $X_1\subset X_2$,
then $H_{X_1}\subset H_{X_2}$.
\varepsilonnd{lemma}
\begin{proposition}f
The former part follows from $X=X+\omega_1$, and this implies the latter part.
\varepsilonnd{proposition}f
\begin{proposition}
For any $\gamma\in{\mathcal G}amma$, we have $\gamma\notin H_{\gamma+\overline{\rm{sg}}(\omega)}$.
\varepsilonnd{proposition}
\begin{proposition}f
By Lemma \ref{H_X}, we have
$$H_{\gamma+\overline{\rm{sg}}(\omega)}=
\bigcap_{n=1}^\infty\overline{\bigcup_{i=n}^\infty(\gamma+\overline{\rm{sg}}(\omega)+\omega_i)}
\subset\overline{\bigcup_{i=2}^\infty(\gamma+\overline{\rm{sg}}(\omega)+\omega_i)}
=\gamma+\overline{\rm{sg}}_1(\omega).$$
Hence $\gamma\notin H_{\gamma+\overline{\rm{sg}}(\omega)}$.
\varepsilonnd{proposition}f
For $\gamma\in{\mathcal G}amma$, we set $P_{\gamma}=I_{\widetilde{X}}$ where
$\widetilde{X}=(\gamma+\overline{\rm{sg}}(\omega),H_{\gamma+\overline{\rm{sg}}(\omega)}\cup\{\gamma\})$
which is a prime $\omega$-invariant pair.
We will show that $P_{\gamma}$ is the unique primitive ideal satisfying that
$\widetilde{X}_{P_{\gamma}}=(\gamma+\overline{\rm{sg}}(\omega),H_{\gamma+\overline{\rm{sg}}(\omega)}\cup\{\gamma\})$.
To see this, we need the following lemma.
\begin{lemma}\label{Pgamma0}
Let $I$ be an ideal of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ with $X_I=\overline{X_I^{(\infty)}+{\rm{sg}}(\omega)}$.
Then $I=I_{\widetilde{X}_I}$.
\varepsilonnd{lemma}
\begin{proposition}f
By the argument in the proof of
Proposition \ref{cond.exp2} and Theorem \ref{idestr1},
it suffices to show that $\|x_0\|\leq\|x\|$ for
$x=\sum_{l=1}^LS_{\mu_l}f_lS_{\nu_l}^*\in ({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I$ and
$x_0=\sum_{|\mu_l|=|\nu_l|}S_{\mu_l}f_lS_{\nu_l}^*$.
If we choose a positive integer $n$
so that $|\mu_l|,|\nu_l|\leq n$ and $\mu_l,\nu_l\in{\mathcal W}_n$ for any $l$,
then $x_0\in {\mathcal F}_I^{(n)}$.
We can find $x_0^{(k)}\in{\mathcal F}_{I}^{(n,k)}\ (0\leq k\leq n-1)$
and $x_0^{(n)}\in{\mathcal G}_{I}^{(n,n)}$ such that $x_0=\sum_{k=0}^nx_0^{(k)}$.
We have $\|x_0\|=\max\{\|x_0^{(0)}\|,\ldots,\|x_0^{(n)}\|\}$.
In the case that $\|x_0\|=\|x_0^{(k)}\|$ for some $k\leq n-1$,
we can prove $\|x_0\|\leq\|x\|$ in a similar way to the proof of
Proposition \ref{cond.exp2}.
In the case that $\|x_0\|=\|x_0^{(n)}\|$,
there exists $\gamma_0\in X_I$
such that $\|x_0\|=\|x_0^{(n)}(\gamma_0)\|$.
Since $X_I=\overline{X_I^{(\infty)}+{\rm{sg}}(\omega)}$,
there exist a sequence $\mu_1,\mu_2,\ldots\in{\mathcal W}_\infty$
and a sequence $\gamma_1,\gamma_2,\ldots,\in X_I^{(\infty)}$
such that $\gamma_0=\lim_{k\to\infty}(\gamma_k+\omega_{\mu_k})$.
We can find sequences $\mu_1',\mu_2',\ldots\in{\mathcal W}_\infty$ and
$\nu_1,\nu_2,\ldots\in{\mathcal W}_n$ such that
$\omega_{\mu_k}=\omega_{\mu_k'}+\omega_{\nu_k}$ and
none of $1,2,\ldots,n$ appears in the word $\mu_k'$ for any $k$.
For $k\in\mathbb{Z}_+$, define a partial isometry
$u_k=\sum_{\mu\in{\mathcal W}_n^{(n)}}S_\mu S_{\nu_k}p_n S_\mu^*$.
We have
$u_k^*xu_k=u_k^*x_0u_k=u_k^*x_0^{(n)}u_k
=\pi_n(\sigma_{\omega_{\nu_k}}x_0^{(n)})$,
where $\pi_n$ is the natural surjection
from ${\mathcal G}_I^{(n,n)}$ onto ${\mathcal F}_I^{(n,n)}$.
Since $\gamma_k\in X_I^{(\infty)}$,
we have $\gamma_k+\omega_{\mu_k'}\in X_I^{(n)}$.
Hence
$$\|\pi_n(\sigma_{\omega_{\nu_k}}x_0^{(n)})\|
\geq\|\sigma_{\omega_{\nu_k}}x_0^{(n)}(\gamma_k+\omega_{\mu_k'})\|
=\|x_0^{(n)}(\gamma_k+\omega_{\mu_k'}+\omega_{\nu_k})\|.$$
Therefore we get
$$\|x_0\|=\|x_0^{(n)}(\gamma_0)\|=\lim_{k\to\infty}
\|x_0^{(n)}(\gamma_k+\omega_{\mu_k'}+\omega_{\nu_k})\|\leq\|x\|.$$
We are done.
\varepsilonnd{proposition}f
| 4,035 | 50,987 |
en
|
train
|
0.36.9
|
\begin{proposition}f
The former part follows from $X=X+\omega_1$, and this implies the latter part.
\varepsilonnd{proposition}f
\begin{proposition}
For any $\gamma\in{\mathcal G}amma$, we have $\gamma\notin H_{\gamma+\overline{\rm{sg}}(\omega)}$.
\varepsilonnd{proposition}
\begin{proposition}f
By Lemma \ref{H_X}, we have
$$H_{\gamma+\overline{\rm{sg}}(\omega)}=
\bigcap_{n=1}^\infty\overline{\bigcup_{i=n}^\infty(\gamma+\overline{\rm{sg}}(\omega)+\omega_i)}
\subset\overline{\bigcup_{i=2}^\infty(\gamma+\overline{\rm{sg}}(\omega)+\omega_i)}
=\gamma+\overline{\rm{sg}}_1(\omega).$$
Hence $\gamma\notin H_{\gamma+\overline{\rm{sg}}(\omega)}$.
\varepsilonnd{proposition}f
For $\gamma\in{\mathcal G}amma$, we set $P_{\gamma}=I_{\widetilde{X}}$ where
$\widetilde{X}=(\gamma+\overline{\rm{sg}}(\omega),H_{\gamma+\overline{\rm{sg}}(\omega)}\cup\{\gamma\})$
which is a prime $\omega$-invariant pair.
We will show that $P_{\gamma}$ is the unique primitive ideal satisfying that
$\widetilde{X}_{P_{\gamma}}=(\gamma+\overline{\rm{sg}}(\omega),H_{\gamma+\overline{\rm{sg}}(\omega)}\cup\{\gamma\})$.
To see this, we need the following lemma.
\begin{lemma}\label{Pgamma0}
Let $I$ be an ideal of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ with $X_I=\overline{X_I^{(\infty)}+{\rm{sg}}(\omega)}$.
Then $I=I_{\widetilde{X}_I}$.
\varepsilonnd{lemma}
\begin{proposition}f
By the argument in the proof of
Proposition \ref{cond.exp2} and Theorem \ref{idestr1},
it suffices to show that $\|x_0\|\leq\|x\|$ for
$x=\sum_{l=1}^LS_{\mu_l}f_lS_{\nu_l}^*\in ({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I$ and
$x_0=\sum_{|\mu_l|=|\nu_l|}S_{\mu_l}f_lS_{\nu_l}^*$.
If we choose a positive integer $n$
so that $|\mu_l|,|\nu_l|\leq n$ and $\mu_l,\nu_l\in{\mathcal W}_n$ for any $l$,
then $x_0\in {\mathcal F}_I^{(n)}$.
We can find $x_0^{(k)}\in{\mathcal F}_{I}^{(n,k)}\ (0\leq k\leq n-1)$
and $x_0^{(n)}\in{\mathcal G}_{I}^{(n,n)}$ such that $x_0=\sum_{k=0}^nx_0^{(k)}$.
We have $\|x_0\|=\max\{\|x_0^{(0)}\|,\ldots,\|x_0^{(n)}\|\}$.
In the case that $\|x_0\|=\|x_0^{(k)}\|$ for some $k\leq n-1$,
we can prove $\|x_0\|\leq\|x\|$ in a similar way to the proof of
Proposition \ref{cond.exp2}.
In the case that $\|x_0\|=\|x_0^{(n)}\|$,
there exists $\gamma_0\in X_I$
such that $\|x_0\|=\|x_0^{(n)}(\gamma_0)\|$.
Since $X_I=\overline{X_I^{(\infty)}+{\rm{sg}}(\omega)}$,
there exist a sequence $\mu_1,\mu_2,\ldots\in{\mathcal W}_\infty$
and a sequence $\gamma_1,\gamma_2,\ldots,\in X_I^{(\infty)}$
such that $\gamma_0=\lim_{k\to\infty}(\gamma_k+\omega_{\mu_k})$.
We can find sequences $\mu_1',\mu_2',\ldots\in{\mathcal W}_\infty$ and
$\nu_1,\nu_2,\ldots\in{\mathcal W}_n$ such that
$\omega_{\mu_k}=\omega_{\mu_k'}+\omega_{\nu_k}$ and
none of $1,2,\ldots,n$ appears in the word $\mu_k'$ for any $k$.
For $k\in\mathbb{Z}_+$, define a partial isometry
$u_k=\sum_{\mu\in{\mathcal W}_n^{(n)}}S_\mu S_{\nu_k}p_n S_\mu^*$.
We have
$u_k^*xu_k=u_k^*x_0u_k=u_k^*x_0^{(n)}u_k
=\pi_n(\sigma_{\omega_{\nu_k}}x_0^{(n)})$,
where $\pi_n$ is the natural surjection
from ${\mathcal G}_I^{(n,n)}$ onto ${\mathcal F}_I^{(n,n)}$.
Since $\gamma_k\in X_I^{(\infty)}$,
we have $\gamma_k+\omega_{\mu_k'}\in X_I^{(n)}$.
Hence
$$\|\pi_n(\sigma_{\omega_{\nu_k}}x_0^{(n)})\|
\geq\|\sigma_{\omega_{\nu_k}}x_0^{(n)}(\gamma_k+\omega_{\mu_k'})\|
=\|x_0^{(n)}(\gamma_k+\omega_{\mu_k'}+\omega_{\nu_k})\|.$$
Therefore we get
$$\|x_0\|=\|x_0^{(n)}(\gamma_0)\|=\lim_{k\to\infty}
\|x_0^{(n)}(\gamma_k+\omega_{\mu_k'}+\omega_{\nu_k})\|\leq\|x\|.$$
We are done.
\varepsilonnd{proposition}f
\begin{proposition}\label{Pgamma}
For any $\gamma\in{\mathcal G}amma$, the ideal $P_{\gamma}$ is the unique primitive
ideal satisfying that
$\widetilde{X}_{P_{\gamma}}=(\gamma+\overline{\rm{sg}}(\omega),H_{\gamma+\overline{\rm{sg}}(\omega)}\cup\{\gamma\})$.
\varepsilonnd{proposition}
\begin{proposition}f
To prove that $P_{\gamma}$ is primitive,
it suffices to show that it is prime because ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ is separable.
Let $I_1,I_2$ be ideals of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$
with $I_1\cap I_2= P_\gamma$.
Then we get
$\widetilde{X}_{I_1}\cup\widetilde{X}_{I_2}=\widetilde{X}_{P_\gamma}$.
Since $\widetilde{X}_{P_\gamma}$ is a prime $\omega$-invariant pair,
we have either
$\widetilde{X}_{I_1}=\widetilde{X}_{P_\gamma}$ or
$\widetilde{X}_{I_2}=\widetilde{X}_{P_\gamma}$.
By Lemma \ref{Pgamma0}, we have either $I_1=P_\gamma$ or $I_2=P_\gamma$.
Therefore $P_\gamma$ is primitive.
The uniqueness follows from Lemma \ref{Pgamma0}.
\varepsilonnd{proposition}f
We denote by $\varDelta$ the set of prime $\omega$-invariant sets
which are not of the form $\gamma+\overline{\rm{sg}}(\omega)$.
For $X\in\varDelta$, we denote by $P_X$ the ideal $I_{\widetilde{X}}$
for $\widetilde{X}=(X,H_X)$ which is a prime $\omega$-invariant pair.
We will show that for any $X\in\varDelta$,
$P_X$ is the unique primitive ideal satisfying $\widetilde{X}_{P_X}=(X,H_X)$.
\begin{lemma}\label{PX0}
Let $X\in\varDelta$ and $\gamma\in X$.
Then there exist a sequence $\mu_1,\mu_2,\ldots$ in ${\mathcal W}_\infty$
and a sequence $\gamma_1,\gamma_2,\ldots$ in $X$ such that
$S_{\mu_k}^*S_1=0$ for any $k$
and $\gamma=\lim_{k\to\infty}(\gamma_k+\omega_{\mu_k})$.
\varepsilonnd{lemma}
\begin{proposition}f
Since $X\in\varDelta$, there exists $\gamma'\in X\setminus (\gamma+\overline{\rm{sg}}(\omega))$.
Since $X$ is prime, Proposition \ref{Xprime} gives us
two sequences $\mu_1,\mu_2,\ldots$, $\nu_1,\nu_2,\ldots$ in ${\mathcal W}_\infty$
and a sequence $\gamma_1,\gamma_2,\ldots$ in $X$ with
$\gamma=\lim_{k\to\infty}(\gamma_k+\omega_{\mu_k})$
and $\gamma'=\lim_{k\to\infty}(\gamma_k+\omega_{\nu_k})$.
We will show that we can choose such $\mu_k$ satisfying $S_{\mu_k}^*S_1=0$.
If not so, then $\mu_k=(1,1,\ldots,1)$ for sufficiently large $k$.
This implies $\gamma'=\lim_{k\to\infty}(\gamma-|\mu_k|\omega_1+\omega_{\nu_k})$
which contradicts the fact that $\gamma'\notin\gamma+\overline{\rm{sg}}(\omega)$.
Therefore we can find desired sequences.
\varepsilonnd{proposition}f
\begin{lemma}\label{PX1}
If an ideal $I$ of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ satisfies $X_I\in\varDelta$,
then $I=I_{\widetilde{X}_I}$.
\varepsilonnd{lemma}
\begin{proposition}f
Similarly as the proof of Lemma \ref{Pgamma0},
it suffices to show that $\|x_0\|\leq\|x\|$ for
$x=\sum_{l=1}^LS_{\mu_l}f_lS_{\nu_l}^*\in ({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I$ and
$x_0=\sum_{|\mu_l|=|\nu_l|}S_{\mu_l}f_lS_{\nu_l}^*\in {\mathcal F}_I^{(n)}$.
We can find $x_0^{(k)}\in{\mathcal F}_{I}^{(n,k)}\ (0\leq k\leq n-1)$
and $x_0^{(n)}\in{\mathcal G}_{I}^{(n,n)}$ such that $x_0=\sum_{k=0}^nx_0^{(k)}$.
We have $\|x_0\|=\max\{\|x_0^{(0)}\|,\ldots,\|x_0^{(n)}\|\}$.
In the case that $\|x_0\|=\|x_0^{(k)}\|$ for some $k\leq n-1$,
we can prove $\|x_0\|\leq\|x\|$ in a similar way to the proof of
Proposition \ref{cond.exp2}.
In the case that $\|x_0\|=\|x_0^{(n)}\|$,
there exists $\gamma_0\in X_I$
such that $\|x_0\|=\|x_0^{(n)}(\gamma_0)\|$.
By Lemma \ref{PX0}, we have a sequence $\mu_1,\mu_2,\ldots$ in ${\mathcal W}_\infty$
and a sequence $\gamma_1,\gamma_2,\ldots$ in $X_I$ such that
$S_{\mu_k}^*S_1=0$ and $\gamma=\lim_{k\to\infty}(\gamma_k+\omega_{\mu_k})$.
For $k\in\mathbb{Z}_+$, set a partial isometry
$u_k=\sum_{\mu\in{\mathcal W}_n^{(n)}}S_\mu S_1^{Kn}S_{\mu_k}S_\mu^*$.
We have
$u_k^*xu_k=u_k^*x_0u_k=u_k^*x_0^{(n)}u_k=\sigma_{\omega_{\mu_k}}x_0^{(n)}$.
Since $\gamma_k\in X_I$, we have
$$\|u_k^*xu_k\|\geq\|\sigma_{\omega_{\mu_k}}x_0^{(n)}(\gamma_k)\|
=\|x_0^{(n)}(\gamma_k+\omega_{\mu_k})\|.$$
Therefore we get
$$\|x_0\|=\|x_0^{(n)}(\gamma)\|=\lim_{k\to\infty}
\|x_0^{(n)}(\gamma_k+\omega_{\mu_k})\|\leq\|x\|.$$
We are done.
\varepsilonnd{proposition}f
\begin{proposition}\label{PX}
For $X\in\varDelta$, the ideal $P_{X}$ is the unique primitive ideal
satisfying $\widetilde{X}_{P_X}=(X,H_X)$.
\varepsilonnd{proposition}
\begin{proposition}f
With the help of Lemma \ref{PX1},
the proof goes similarly as the one in Proposition \ref{Pgamma}.
\varepsilonnd{proposition}f
By Proposition \ref{pp}, the remaining candidates for primitive ideals
are ideals $P$ satisfying
$\widetilde{X}_P=(\gamma_0+\overline{\rm{sg}}(\omega),H_{\gamma_0+\overline{\rm{sg}}(\omega)})$
for some $\gamma_0\in{\mathcal G}amma$.
We will determine such primitive ideals.
\begin{definition}\rm\label{DefP}
For $([\gamma],\theta)\in{\mathcal G}amma'\times\mathbb{T}$,
we set $Y_{([\gamma],\theta)}=\big\{([\gamma],\theta)\big\}\cup
\big([\gamma+\overline{\rm{sg}}_1(\omega)]\times\mathbb{T}\big)$.
Then $\widetilde{Y}=(Y_{([\gamma],\theta)},H_{\gamma+\overline{\rm{sg}}(\omega)})$
is an $\omega$-invariant pair.
We write $P_{([\gamma],\theta)}$
for denoting $I_{\widetilde{Y}}$.
\varepsilonnd{definition}
We can show that $P_{([\gamma],\theta)}$ is a primitive ideal
for any $([\gamma],\theta)\in{\mathcal G}amma'\times\mathbb{T}$
by using the technique in \cite{Ka1}.
To do so, we need Proposition \ref{local}, which
will also be used to determine the topology of primitive ideal space of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$.
\begin{lemma}\label{I_X=}
For an $\omega$-invariant set $X$,
the pair $\widetilde{X}=(X,X)$ is $\omega$-invariant and we have
$$I_{\widetilde{X}}=\cspa\{S_\mu fS_\nu^*\mid
\mu,\nu\in{\mathcal W}_\infty,\ f\in C_0({\mathcal G}amma\setminus X)\}.$$
\varepsilonnd{lemma}
\begin{proposition}f
Clearly, $\widetilde{X}=(X,X)$ is $\omega$-invariant.
Set $I=\cspa\{S_\mu fS_\nu^*\mid
\mu,\nu\in{\mathcal W}_\infty,\ f\in C_0({\mathcal G}amma\setminus X)\}$.
In a similar way to the proof of Proposition \ref{I_X,Xinfty},
we can see that $I$ is a gauge-invariant ideal of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$.
We also see that $X_I^{(n)}=X$ for any $n\in\mathbb{N}$
by arguing as in the proof of Proposition \ref{exist}.
Hence $I_{\widetilde{X}}=I$ by Theorem \ref{OneToOne}.
\varepsilonnd{proposition}f
| 3,923 | 50,987 |
en
|
train
|
0.36.10
|
\begin{proposition}f
Similarly as the proof of Lemma \ref{Pgamma0},
it suffices to show that $\|x_0\|\leq\|x\|$ for
$x=\sum_{l=1}^LS_{\mu_l}f_lS_{\nu_l}^*\in ({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I$ and
$x_0=\sum_{|\mu_l|=|\nu_l|}S_{\mu_l}f_lS_{\nu_l}^*\in {\mathcal F}_I^{(n)}$.
We can find $x_0^{(k)}\in{\mathcal F}_{I}^{(n,k)}\ (0\leq k\leq n-1)$
and $x_0^{(n)}\in{\mathcal G}_{I}^{(n,n)}$ such that $x_0=\sum_{k=0}^nx_0^{(k)}$.
We have $\|x_0\|=\max\{\|x_0^{(0)}\|,\ldots,\|x_0^{(n)}\|\}$.
In the case that $\|x_0\|=\|x_0^{(k)}\|$ for some $k\leq n-1$,
we can prove $\|x_0\|\leq\|x\|$ in a similar way to the proof of
Proposition \ref{cond.exp2}.
In the case that $\|x_0\|=\|x_0^{(n)}\|$,
there exists $\gamma_0\in X_I$
such that $\|x_0\|=\|x_0^{(n)}(\gamma_0)\|$.
By Lemma \ref{PX0}, we have a sequence $\mu_1,\mu_2,\ldots$ in ${\mathcal W}_\infty$
and a sequence $\gamma_1,\gamma_2,\ldots$ in $X_I$ such that
$S_{\mu_k}^*S_1=0$ and $\gamma=\lim_{k\to\infty}(\gamma_k+\omega_{\mu_k})$.
For $k\in\mathbb{Z}_+$, set a partial isometry
$u_k=\sum_{\mu\in{\mathcal W}_n^{(n)}}S_\mu S_1^{Kn}S_{\mu_k}S_\mu^*$.
We have
$u_k^*xu_k=u_k^*x_0u_k=u_k^*x_0^{(n)}u_k=\sigma_{\omega_{\mu_k}}x_0^{(n)}$.
Since $\gamma_k\in X_I$, we have
$$\|u_k^*xu_k\|\geq\|\sigma_{\omega_{\mu_k}}x_0^{(n)}(\gamma_k)\|
=\|x_0^{(n)}(\gamma_k+\omega_{\mu_k})\|.$$
Therefore we get
$$\|x_0\|=\|x_0^{(n)}(\gamma)\|=\lim_{k\to\infty}
\|x_0^{(n)}(\gamma_k+\omega_{\mu_k})\|\leq\|x\|.$$
We are done.
\varepsilonnd{proposition}f
\begin{proposition}\label{PX}
For $X\in\varDelta$, the ideal $P_{X}$ is the unique primitive ideal
satisfying $\widetilde{X}_{P_X}=(X,H_X)$.
\varepsilonnd{proposition}
\begin{proposition}f
With the help of Lemma \ref{PX1},
the proof goes similarly as the one in Proposition \ref{Pgamma}.
\varepsilonnd{proposition}f
By Proposition \ref{pp}, the remaining candidates for primitive ideals
are ideals $P$ satisfying
$\widetilde{X}_P=(\gamma_0+\overline{\rm{sg}}(\omega),H_{\gamma_0+\overline{\rm{sg}}(\omega)})$
for some $\gamma_0\in{\mathcal G}amma$.
We will determine such primitive ideals.
\begin{definition}\rm\label{DefP}
For $([\gamma],\theta)\in{\mathcal G}amma'\times\mathbb{T}$,
we set $Y_{([\gamma],\theta)}=\big\{([\gamma],\theta)\big\}\cup
\big([\gamma+\overline{\rm{sg}}_1(\omega)]\times\mathbb{T}\big)$.
Then $\widetilde{Y}=(Y_{([\gamma],\theta)},H_{\gamma+\overline{\rm{sg}}(\omega)})$
is an $\omega$-invariant pair.
We write $P_{([\gamma],\theta)}$
for denoting $I_{\widetilde{Y}}$.
\varepsilonnd{definition}
We can show that $P_{([\gamma],\theta)}$ is a primitive ideal
for any $([\gamma],\theta)\in{\mathcal G}amma'\times\mathbb{T}$
by using the technique in \cite{Ka1}.
To do so, we need Proposition \ref{local}, which
will also be used to determine the topology of primitive ideal space of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$.
\begin{lemma}\label{I_X=}
For an $\omega$-invariant set $X$,
the pair $\widetilde{X}=(X,X)$ is $\omega$-invariant and we have
$$I_{\widetilde{X}}=\cspa\{S_\mu fS_\nu^*\mid
\mu,\nu\in{\mathcal W}_\infty,\ f\in C_0({\mathcal G}amma\setminus X)\}.$$
\varepsilonnd{lemma}
\begin{proposition}f
Clearly, $\widetilde{X}=(X,X)$ is $\omega$-invariant.
Set $I=\cspa\{S_\mu fS_\nu^*\mid
\mu,\nu\in{\mathcal W}_\infty,\ f\in C_0({\mathcal G}amma\setminus X)\}$.
In a similar way to the proof of Proposition \ref{I_X,Xinfty},
we can see that $I$ is a gauge-invariant ideal of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$.
We also see that $X_I^{(n)}=X$ for any $n\in\mathbb{N}$
by arguing as in the proof of Proposition \ref{exist}.
Hence $I_{\widetilde{X}}=I$ by Theorem \ref{OneToOne}.
\varepsilonnd{proposition}f
\begin{proposition}\label{local}
Let $X$ be a compact subset of ${\mathcal G}amma$ such that
$X\cap (X+\gamma)=\varepsilonmptyset$ for any $\gamma\in\overline{\rm{sg}}(\omega)\setminus\{0\}$,
and set $X_1=X+\overline{\rm{sg}}(\omega)$ and $X_2=X+\overline{\rm{sg}}_1(\omega)$.
Then we have that $\widetilde{X}_0=(X_1,X_1)$, $\widetilde{X}_1=(X_1,X_2)$ and
$\widetilde{X}_2=(X_2,X_2)$ are $\omega$-invariant pairs,
and that
\begin{align*}
I_{\widetilde{X}_2}/I_{\widetilde{X}_1}&\cong\mathbb{K}\otimes C(X\times\mathbb{T}),&
I_{\widetilde{X}_1}/I_{\widetilde{X}_0}&\cong\mathbb{K}\otimes C(X_1\setminus X_2).
\varepsilonnd{align*}
\varepsilonnd{proposition}
\begin{proposition}f
Since $X$ is compact and $\overline{\rm{sg}}(\omega)$ is closed, $X_1=X+\overline{\rm{sg}}(\omega)$ becomes closed.
The same reason shows that $X_2$ is closed.
By Lemma \ref{isolate}, both $X_1$ and $X_2$ are $\omega$-invariant and
$X_2=\overline{\bigcup_{i=2}^\infty(X_1+\omega_i)}$.
Therefore $\widetilde{X}_0,\widetilde{X}_1,\widetilde{X}_2$ are
$\omega$-invariant pairs.
Since $I_{\widetilde{X}_1}\cap p_1 C_0({\mathcal G}amma)=p_1 C_0({\mathcal G}amma\setminus X_2)$,
we have $p_1f=0$
for any $f\in C_0(X_1\setminus X_2)\subset I_{\widetilde{X}_2}/I_{\widetilde{X}_1}$.
Note that $X_1\setminus X_2$ is a disjoint union of compact sets
$X,X+\omega_1,\ldots,X+(K-1)\omega_1$.
For $f\in C(X+m\omega_1)\subset I_{\widetilde{X}_2}/I_{\widetilde{X}_1}$
with $0<m<K$,
we have $\sigma_{m\omega_1}f\in C(X)$ and
\begin{align*}
S_1^m\sigma_{m\omega_1}f{S_1^*}^m
&=S_1^{m-1}S_1S_1^*\sigma_{(m-1)\omega_1}f{S_1^*}^{m-1}
=S_1^{m-1}\sigma_{(m-1)\omega_1}f{S_1^*}^{m-1}\\
&=\cdots=f.
\varepsilonnd{align*}
Hence, we have
$I_{\widetilde{X}_2}/I_{\widetilde{X}_1}
=\cspa\{S_\mu fS_\nu^*\mid \mu,\nu\in{\mathcal W}_\infty, f\in C(X)\}$
by Lemma \ref{I_X=}.
Set ${\mathcal W}_\infty^+={\mathcal W}_\infty\setminus\{\mu 1^K\in{\mathcal W}_\infty\mid\mu\in{\mathcal W}_\infty\}$
and denote by $\chi$ the characteristic function of $X$.
Then $\{S_\mu\chi S_\nu^*\}_{\mu,\nu\in {\mathcal W}_\infty^+}$
satisfies the relation of matrix units
and $\sum_{\mu\in{\mathcal W}_\infty^+}S_\mu\chi S_\mu^*=1$
(strictly).
Hence we have $I_{\widetilde{X}_2}/I_{\widetilde{X}_1}\cong\mathbb{K}\otimes B$
where $B=\chi(I_{\widetilde{X}_2}/I_{\widetilde{X}_1})\chi$.
We have
$$B=\cspa\{\chi S_\mu fS_\nu^* \chi\mid \mu,\nu\in{\mathcal W}_\infty, f\in C(X)\}
=\cspa\{(S_1^K)^m f\mid m\in\mathbb{Z}, f\in C(X)\}.$$
Since $B$ is generated by $C(X)$ and a unitary $S_1^K\chi$ which commute
with each other and since $B$ is globally invariant under the gauge action,
we have $B\cong C(X\times\mathbb{T})$.
Therefore we get
$I_{\widetilde{X}_2}/I_{\widetilde{X}_1}\cong\mathbb{K}\otimes C(X\times\mathbb{T})$.
By the definition,
$$I_{\widetilde{X}_1}/I_{\widetilde{X}_0}
=\cspa\{S_\mu p_nfS_\nu^*\mid \mu,\nu\in{\mathcal W}_\infty, n\geq 1,\
f\in C(X_1\setminus X_2)\}.$$
For $f\in C(X_1\setminus X_2)\subset
I_{\widetilde{X}_1}/I_{\widetilde{X}_0}$ and $i\geq 2$,
we have $S_iS_i^*f=S_i\sigma_{\omega_i}fS_i^*=0$.
Hence $p_nf=p_1f$ for any $n\geq 1$
and any $f\in C(X_1\setminus X_2)$.
Thus $I_{X_1,X_2}/I_{X_1,X_1}
=\cspa\{S_\mu p_2fS_\nu^*\mid \mu,\nu\in{\mathcal W}_\infty,\
f\in C(X_1\setminus X_2)\}.$
We can show that
$\{S_\mu p_2\chi'S_\nu^*\}_{\mu,\nu\in{\mathcal W}_\infty}$
satisfies the relation of matrix units
and $\sum_{\mu\in{\mathcal W}_\infty}S_\mu p_2\chi'S_\mu^*=1$ (strictly),
where $\chi'$ is the characteristic function of $X_1\setminus X_2$.
Hence we have $I_{\widetilde{X}_1}/I_{\widetilde{X}_0}\cong\mathbb{K}\otimes B'$
where
$$B'=p_2\chi'(I_{\widetilde{X}_1}/I_{\widetilde{X}_0})p_2\chi'
=\cspa\{p_2f\mid f\in C(X_1\setminus X_2)\}\cong C(X_1\setminus X_2).$$
Therefore we get
$I_{\widetilde{X}_1}/I_{\widetilde{X}_0}\cong\mathbb{K}\otimes C(X_1\setminus X_2)$.
\varepsilonnd{proposition}f
With the help of Proposition \ref{local},
we have the following proposition
by exactly the same argument as the proof of \cite[Proposition 5.41]{Ka1}.
\begin{proposition}\label{Primsurj}
For $\gamma_0\in{\mathcal G}amma$, the set of all primitive ideals $P$ satisfying
$\widetilde{X}_P=(\gamma_0+\overline{\rm{sg}}(\omega),H_{\gamma_0+\overline{\rm{sg}}(\omega)})$
is $\{P_{([\gamma_0],\theta)}\mid\theta\in\mathbb{T}\}$.
\varepsilonnd{proposition}
Now, we can describe the primitive ideal space $\Prim({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)$ of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$
as follows.
\begin{proposition}\label{primitive2}
We have
$\Prim({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)=\{P_z\mid z\in ({\mathcal G}amma'\times\mathbb{T})\sqcup{\mathcal G}amma\sqcup\varDelta\}$,
where $\sqcup$ means a disjoint union.
\varepsilonnd{proposition}
The primitive ideal space $\Prim({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)$ is a topological space
whose closed sets are given by
$\{P\in\Prim({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)\mid I\subset P\}$ for ideals $I$.
We will investigate which subset of
$({\mathcal G}amma'\times\mathbb{T})\sqcup{\mathcal G}amma\sqcup\varDelta$
corresponds to a closed subset of $\Prim({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)$.
By Corollary \ref{Ysub}, the following is easy to verify.
\begin{lemma}\label{Primlem1}
Let $\widetilde{Y}=(Y,X^{(\infty)})$ be an $\omega$-invariant set.
\benu
\item For $([\gamma],\theta)\in{\mathcal G}amma'\times\mathbb{T}$, we have
$I_{\widetilde{Y}}\subset P_{([\gamma],\theta)}$ if and only if
$([\gamma],\theta)\in Y$.
\item For $\gamma\in{\mathcal G}amma$, we have
$I_{\widetilde{Y}}\subset P_{\gamma}$ if and only if
$\gamma\in X^{(\infty)}$.
\item For $X\in\varDelta$, we have
$I_{\widetilde{Y}}\subset P_{X}$ if and only if
$[X]\times\mathbb{T}\subset Y$.
\varepsilonnd{enumerate}
\varepsilonnd{lemma}
\begin{lemma}\label{Primlem2}
Let $X$ be a compact subset of ${\mathcal G}amma$
such that $X\cap (X+\gamma)=\varepsilonmptyset$ for any $\gamma\in\overline{\rm{sg}}(\omega)\setminus\{0\}$,
and set $X_1=X+\overline{\rm{sg}}(\omega)$ and $X_2=X+\overline{\rm{sg}}_1(\omega)$, which are $\omega$-invariant sets.
If $X_0\in\varDelta$ satisfies $X_1\supset X_0$, then $X_2\supset X_0$.
\varepsilonnd{lemma}
| 3,994 | 50,987 |
en
|
train
|
0.36.11
|
By the definition,
$$I_{\widetilde{X}_1}/I_{\widetilde{X}_0}
=\cspa\{S_\mu p_nfS_\nu^*\mid \mu,\nu\in{\mathcal W}_\infty, n\geq 1,\
f\in C(X_1\setminus X_2)\}.$$
For $f\in C(X_1\setminus X_2)\subset
I_{\widetilde{X}_1}/I_{\widetilde{X}_0}$ and $i\geq 2$,
we have $S_iS_i^*f=S_i\sigma_{\omega_i}fS_i^*=0$.
Hence $p_nf=p_1f$ for any $n\geq 1$
and any $f\in C(X_1\setminus X_2)$.
Thus $I_{X_1,X_2}/I_{X_1,X_1}
=\cspa\{S_\mu p_2fS_\nu^*\mid \mu,\nu\in{\mathcal W}_\infty,\
f\in C(X_1\setminus X_2)\}.$
We can show that
$\{S_\mu p_2\chi'S_\nu^*\}_{\mu,\nu\in{\mathcal W}_\infty}$
satisfies the relation of matrix units
and $\sum_{\mu\in{\mathcal W}_\infty}S_\mu p_2\chi'S_\mu^*=1$ (strictly),
where $\chi'$ is the characteristic function of $X_1\setminus X_2$.
Hence we have $I_{\widetilde{X}_1}/I_{\widetilde{X}_0}\cong\mathbb{K}\otimes B'$
where
$$B'=p_2\chi'(I_{\widetilde{X}_1}/I_{\widetilde{X}_0})p_2\chi'
=\cspa\{p_2f\mid f\in C(X_1\setminus X_2)\}\cong C(X_1\setminus X_2).$$
Therefore we get
$I_{\widetilde{X}_1}/I_{\widetilde{X}_0}\cong\mathbb{K}\otimes C(X_1\setminus X_2)$.
\varepsilonnd{proposition}f
With the help of Proposition \ref{local},
we have the following proposition
by exactly the same argument as the proof of \cite[Proposition 5.41]{Ka1}.
\begin{proposition}\label{Primsurj}
For $\gamma_0\in{\mathcal G}amma$, the set of all primitive ideals $P$ satisfying
$\widetilde{X}_P=(\gamma_0+\overline{\rm{sg}}(\omega),H_{\gamma_0+\overline{\rm{sg}}(\omega)})$
is $\{P_{([\gamma_0],\theta)}\mid\theta\in\mathbb{T}\}$.
\varepsilonnd{proposition}
Now, we can describe the primitive ideal space $\Prim({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)$ of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$
as follows.
\begin{proposition}\label{primitive2}
We have
$\Prim({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)=\{P_z\mid z\in ({\mathcal G}amma'\times\mathbb{T})\sqcup{\mathcal G}amma\sqcup\varDelta\}$,
where $\sqcup$ means a disjoint union.
\varepsilonnd{proposition}
The primitive ideal space $\Prim({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)$ is a topological space
whose closed sets are given by
$\{P\in\Prim({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)\mid I\subset P\}$ for ideals $I$.
We will investigate which subset of
$({\mathcal G}amma'\times\mathbb{T})\sqcup{\mathcal G}amma\sqcup\varDelta$
corresponds to a closed subset of $\Prim({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)$.
By Corollary \ref{Ysub}, the following is easy to verify.
\begin{lemma}\label{Primlem1}
Let $\widetilde{Y}=(Y,X^{(\infty)})$ be an $\omega$-invariant set.
\benu
\item For $([\gamma],\theta)\in{\mathcal G}amma'\times\mathbb{T}$, we have
$I_{\widetilde{Y}}\subset P_{([\gamma],\theta)}$ if and only if
$([\gamma],\theta)\in Y$.
\item For $\gamma\in{\mathcal G}amma$, we have
$I_{\widetilde{Y}}\subset P_{\gamma}$ if and only if
$\gamma\in X^{(\infty)}$.
\item For $X\in\varDelta$, we have
$I_{\widetilde{Y}}\subset P_{X}$ if and only if
$[X]\times\mathbb{T}\subset Y$.
\varepsilonnd{enumerate}
\varepsilonnd{lemma}
\begin{lemma}\label{Primlem2}
Let $X$ be a compact subset of ${\mathcal G}amma$
such that $X\cap (X+\gamma)=\varepsilonmptyset$ for any $\gamma\in\overline{\rm{sg}}(\omega)\setminus\{0\}$,
and set $X_1=X+\overline{\rm{sg}}(\omega)$ and $X_2=X+\overline{\rm{sg}}_1(\omega)$, which are $\omega$-invariant sets.
If $X_0\in\varDelta$ satisfies $X_1\supset X_0$, then $X_2\supset X_0$.
\varepsilonnd{lemma}
\begin{proposition}f
To the contrary, assume $X_0\in\varDelta$ satisfies
$X_1\supset X_0$ and $X_2\not\supset X_0$.
Then $X_0\cap X\neq\varepsilonmptyset$ and
$(X_0\cap X)+\overline{\rm{sg}}(\omega)$ is an $\omega$-invariant set
satisfying $(X_0\cap X)+\overline{\rm{sg}}(\omega)\subset X_0$.
Since $((X_0\cap X)+\overline{\rm{sg}}(\omega)) \cup X_2\supset X_0$ and $X_0$ is prime,
we have $(X_0\cap X)+\overline{\rm{sg}}(\omega)\supset X_0$.
Hence $X_0=(X_0\cap X)+\overline{\rm{sg}}(\omega)$.
If $X_0\cap X$ has two points $\gamma_1,\gamma_2$,
then we can take open sets $U_1,U_2$ such that $\gamma_1\in U_1$,
$\gamma_2\in U_2$, $U_1\cap U_2=\varepsilonmptyset$.
Two $\omega$-invariant sets $X_1'=(X_0\cap X\setminus U_1)+\overline{\rm{sg}}(\omega)$,
$X_2'=(X_0\cap X\setminus U_2)+\overline{\rm{sg}}(\omega)$ satisfies
$X_1'\not\supset X_0$, $X_2'\not\supset X_0$ and $X_1'\cup X_2'=X_0$.
This contradicts the primeness of $X_0$.
Hence $X_0\cap X$ is just a point.
However, this contradicts the fact that $X_0\in\varDelta$.
Therefore $X_2\supset X_0$ when $X_0\in\varDelta$ satisfies $X_1\supset X_0$.
\varepsilonnd{proposition}f
\begin{lemma}\label{Primlem3}
Let $\widetilde{Y}_\lambda=(Y_\lambda,X_\lambda^{(\infty)})$
be an $\omega$-invariant pair for each $\lambda\in\Lambda$.
Set $I=\bigcap_{\lambda\in\Lambda}I_{\widetilde{Y}_\lambda}$.
Then $Y_I=\overline{\bigcup_{\lambda\in\Lambda}Y_\lambda}$.
\varepsilonnd{lemma}
\begin{proposition}f
For any $\lambda\in\Lambda$, we have $Y_I\supset Y_\lambda$
because $I\subset I_{\widetilde{Y}_\lambda}$.
Hence we get $Y_I\supset \overline{\bigcup_{\lambda\in\Lambda}Y_\lambda}$.
Take
$([\gamma_0],\theta_0)\notin
\overline{\bigcup_{\lambda\in\Lambda}Y_\lambda}$.
Then
there exists a neighborhood $U$ of $([\gamma_0],\theta_0)$ satisfying
$U\cap\overline{\bigcup_{\lambda\in\Lambda}Y_\lambda}=\varepsilonmptyset$.
By the same argument as in the proof of \cite[Lemma 5.22]{Ka1},
we can find $x_0\in A$ such that
$\psi_{([\gamma_0],\theta_0)}(x_0)\neq 0$ and
$\psi_{([\gamma],\theta)}(x_0)=0$ if $([\gamma],\theta)\notin U$
and $\varphi_{\gamma}(x_0)=0$ if $([\gamma]\times\mathbb{T})\cap U=\varepsilonmptyset$.
Therefore we have $x_0\in I$,
and it implies that
$([\gamma_0],\theta_0)\notin Y_I$.
Thus $Y_I=\overline{\bigcup_{\lambda\in\Lambda}Y_\lambda}$.
\varepsilonnd{proposition}f
\begin{lemma}\label{Primlem4}
For any $X\in\varDelta$, we have $P_{X}=\bigcap_{z\in [X]\times\mathbb{T}}P_z$.
\varepsilonnd{lemma}
\begin{proposition}f
By Lemma \ref{Primlem1},
we have $P_{X}\subset\bigcap_{z\in [X]\times\mathbb{T}}P_z$.
By Lemma \ref{Primlem3},
we have $Y_{\bigcap_{z\in [X]\times\mathbb{T}}P_z}=[X]\times\mathbb{T}$.
Hence we have $\bigcap_{z\in [X]\times\mathbb{T}}P_z\subset P_{X}$ by
Lemma \ref{Primlem1}.
Thus $P_{X}=\bigcap_{z\in [X]\times\mathbb{T}}P_z$.
\varepsilonnd{proposition}f
In the proof of the following proposition,
we use the fact that
the subset $\{P\in\Prim({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)\mid I_1\subset P, I_2\not\subset P\}$
of $\Prim({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)$ is homeomorphic to $\Prim(I_2/I_1)$,
for two ideals $I_1,I_2$ of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ with $I_1\subset I_2$.
\begin{proposition}\label{closed}
Let $Z=Y\sqcup X^{(\infty)}\sqcup \varLambda$ be a subset of
$({\mathcal G}amma'\times\mathbb{T})\sqcup{\mathcal G}amma\sqcup\varDelta$.
The set $P_Z=\{P_z\mid z\in Z\}$ is closed in $\Prim({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)$
if and only if $(Y,X^{(\infty)})$ is an $\omega$-invariant set and
$\varLambda=\{X\in\varDelta\mid [X]\times\mathbb{T}\subset Y\}$.
\varepsilonnd{proposition}
\begin{proposition}f
Let us take a subset $Z=Y\sqcup X^{(\infty)}\sqcup \varLambda$ of
$({\mathcal G}amma'\times\mathbb{T})\sqcup{\mathcal G}amma\sqcup\varDelta$
satisfying that $(Y,X^{(\infty)})$ is an $\omega$-invariant set and
$\varLambda=\{X\in\varDelta\mid [X]\times\mathbb{T}\subset Y\}$.
Then the set $P_Z=\{P_z\mid z\in Z\}$ coincides
with the closed subset defined by the ideal $I_{\widetilde{Y}}$
by Lemma \ref{Primlem1}.
| 2,969 | 50,987 |
en
|
train
|
0.36.12
|
\begin{lemma}\label{Primlem3}
Let $\widetilde{Y}_\lambda=(Y_\lambda,X_\lambda^{(\infty)})$
be an $\omega$-invariant pair for each $\lambda\in\Lambda$.
Set $I=\bigcap_{\lambda\in\Lambda}I_{\widetilde{Y}_\lambda}$.
Then $Y_I=\overline{\bigcup_{\lambda\in\Lambda}Y_\lambda}$.
\varepsilonnd{lemma}
\begin{proposition}f
For any $\lambda\in\Lambda$, we have $Y_I\supset Y_\lambda$
because $I\subset I_{\widetilde{Y}_\lambda}$.
Hence we get $Y_I\supset \overline{\bigcup_{\lambda\in\Lambda}Y_\lambda}$.
Take
$([\gamma_0],\theta_0)\notin
\overline{\bigcup_{\lambda\in\Lambda}Y_\lambda}$.
Then
there exists a neighborhood $U$ of $([\gamma_0],\theta_0)$ satisfying
$U\cap\overline{\bigcup_{\lambda\in\Lambda}Y_\lambda}=\varepsilonmptyset$.
By the same argument as in the proof of \cite[Lemma 5.22]{Ka1},
we can find $x_0\in A$ such that
$\psi_{([\gamma_0],\theta_0)}(x_0)\neq 0$ and
$\psi_{([\gamma],\theta)}(x_0)=0$ if $([\gamma],\theta)\notin U$
and $\varphi_{\gamma}(x_0)=0$ if $([\gamma]\times\mathbb{T})\cap U=\varepsilonmptyset$.
Therefore we have $x_0\in I$,
and it implies that
$([\gamma_0],\theta_0)\notin Y_I$.
Thus $Y_I=\overline{\bigcup_{\lambda\in\Lambda}Y_\lambda}$.
\varepsilonnd{proposition}f
\begin{lemma}\label{Primlem4}
For any $X\in\varDelta$, we have $P_{X}=\bigcap_{z\in [X]\times\mathbb{T}}P_z$.
\varepsilonnd{lemma}
\begin{proposition}f
By Lemma \ref{Primlem1},
we have $P_{X}\subset\bigcap_{z\in [X]\times\mathbb{T}}P_z$.
By Lemma \ref{Primlem3},
we have $Y_{\bigcap_{z\in [X]\times\mathbb{T}}P_z}=[X]\times\mathbb{T}$.
Hence we have $\bigcap_{z\in [X]\times\mathbb{T}}P_z\subset P_{X}$ by
Lemma \ref{Primlem1}.
Thus $P_{X}=\bigcap_{z\in [X]\times\mathbb{T}}P_z$.
\varepsilonnd{proposition}f
In the proof of the following proposition,
we use the fact that
the subset $\{P\in\Prim({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)\mid I_1\subset P, I_2\not\subset P\}$
of $\Prim({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)$ is homeomorphic to $\Prim(I_2/I_1)$,
for two ideals $I_1,I_2$ of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ with $I_1\subset I_2$.
\begin{proposition}\label{closed}
Let $Z=Y\sqcup X^{(\infty)}\sqcup \varLambda$ be a subset of
$({\mathcal G}amma'\times\mathbb{T})\sqcup{\mathcal G}amma\sqcup\varDelta$.
The set $P_Z=\{P_z\mid z\in Z\}$ is closed in $\Prim({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)$
if and only if $(Y,X^{(\infty)})$ is an $\omega$-invariant set and
$\varLambda=\{X\in\varDelta\mid [X]\times\mathbb{T}\subset Y\}$.
\varepsilonnd{proposition}
\begin{proposition}f
Let us take a subset $Z=Y\sqcup X^{(\infty)}\sqcup \varLambda$ of
$({\mathcal G}amma'\times\mathbb{T})\sqcup{\mathcal G}amma\sqcup\varDelta$
satisfying that $(Y,X^{(\infty)})$ is an $\omega$-invariant set and
$\varLambda=\{X\in\varDelta\mid [X]\times\mathbb{T}\subset Y\}$.
Then the set $P_Z=\{P_z\mid z\in Z\}$ coincides
with the closed subset defined by the ideal $I_{\widetilde{Y}}$
by Lemma \ref{Primlem1}.
Conversely, assume $P_Z$ is closed, that is,
there exists an ideal $I$ of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ with
$Z=\{z\in Y\sqcup X^{(\infty)}\sqcup \varLambda\mid I\subset P_z\}$.
We first show that $Y$ and $X^{(\infty)}$ is closed.
Take $\gamma_0\in{\mathcal G}amma$ arbitrarily.
By Corollary \ref{cptnbhd},
there exists a compact neighborhood $X$ of $\gamma_0$
such that $X\cap (X+\gamma)=\varepsilonmptyset$ for any $\gamma\in\overline{\rm{sg}}(\omega)\setminus\{0\}$.
Set $\widetilde{X}_0=(X_1,X_1)$, $\widetilde{X}_1=(X_1,X_2)$ and
$\widetilde{X}_2=(X_2,X_2)$ where $X_1=X+\overline{\rm{sg}}(\omega)$ and $X_2=X+\overline{\rm{sg}}_1(\omega)$.
Note that $X\ni\gamma\mapsto[\gamma]\in [X_1\setminus X_2]$ is a homeomorphism.
By Lemma \ref{Primlem1} and Lemma \ref{Primlem2}, we have
\begin{align*}
\big\{z\in ({\mathcal G}amma'\times\mathbb{T})\sqcup{\mathcal G}amma\sqcup\varDelta\mid
I_{\widetilde{X}_1}\subset P_z, I_{\widetilde{X}_2}\not\subset P_z\big\}
&=[X_1\setminus X_2]\times\mathbb{T}\subset {\mathcal G}amma'\times\mathbb{T},\\
\big\{z\in ({\mathcal G}amma'\times\mathbb{T})\sqcup{\mathcal G}amma\sqcup\varDelta\mid
I_{\widetilde{X}_0}\subset P_z, I_{\widetilde{X}_1}\not\subset P_z\big\}
&=X_1\setminus X_2\subset{\mathcal G}amma.
\varepsilonnd{align*}
By Proposition \ref{local}, the map
$[X_1\setminus X_2]\times\mathbb{T}\ni z\mapsto P_z\in\Prim({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)$ is a homeomorphism
from $[X_1\setminus X_2]\times\mathbb{T}$,
whose topology is the relative topology of ${\mathcal G}amma'\times\mathbb{T}$,
to the subset
$\{P\in\Prim({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)\mid I_{\widetilde{X}_1}\subset P,
I_{\widetilde{X}_2}\not\subset P\}$
of $\Prim({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)$.
The set $Y\cap ([X_1\setminus X_2]\times\mathbb{T})\subset {\mathcal G}amma'\times\mathbb{T}$ is closed
in $[X_1\setminus X_2]\times\mathbb{T}$ because $P_Y$ is closed.
Hence, the subset $Y$ is closed in ${\mathcal G}amma'\times\mathbb{T}$.
Similarly $X^{(\infty)}$ is closed in ${\mathcal G}amma$.
Set $X=\{\gamma\in{\mathcal G}amma\mid ([\gamma],\theta)\in Y \mbox{ for some }\theta\in\mathbb{T}\}$, which is closed because $Y$ is closed.
Set $J=\bigcap_{([\gamma],\theta)\in Y}P_{([\gamma],\theta)}$.
We have $I\subset J$.
By Lemma \ref{Primlem3}, we have $Y_J=Y$.
Hence $H_X\subset X_J^{(\infty)}$.
We have $J\subset P_\gamma$ for any $\gamma\in H_X$ by Lemma \ref{Primlem1}.
Therefore $H_X\subset X^{(\infty)}$.
We have $([\gamma+\omega_i],\theta')\in Y$
for any $([\gamma],\theta)\in Y$, any $i\geq 2$ and any $\theta'\in\mathbb{T}$
because $P_{([\gamma],\theta)}\subset P_{([\gamma+\omega_i],\theta')}$.
Hence we get $[X+\omega_i]\times\mathbb{T}\subset Y$.
We also have $[X^{(\infty)}]\times\mathbb{T}\subset Y$
because $P_\gamma\subset P_{([\gamma],\theta)}$
for any $([\gamma],\theta)\in{\mathcal G}amma'\times\mathbb{T}$.
Therefore we have proved that $(Y,X^{(\infty)})$ is an $\omega$-invariant set.
Finally, we have $\varLambda=\{X\in\varDelta\mid [X]\times\mathbb{T}\subset Y\}$
by Lemma \ref{Primlem4}.
It completes the proof.
\varepsilonnd{proposition}f
By the proposition above, we get the following.
\begin{theorem}\label{idestr2}
When $\omega$ does not satisfy Condition \ref{cond},
there is a one-to-one correspondence between the set of ideals of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$
and the set of $\omega$-invariant pairs of subsets of ${\mathcal G}amma'\times\mathbb{T}$
and subsets of ${\mathcal G}amma$.
Hence for any ideal $I$ of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$, we have $I=I_{\widetilde{Y}_I}$.
\varepsilonnd{theorem}
\begin{proposition}f
There is a one-to-one correspondence
between the set of ideals of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ and the closed subset of $\Prim({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)$.
By Proposition \ref{closed},
the closed subset of $\Prim({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)$ corresponds bijectively
to the set of $\omega$-invariant pairs.
\varepsilonnd{proposition}f
| 2,628 | 50,987 |
en
|
train
|
0.36.13
|
\section{More about ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$}
In this section, we gather some general results on ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$.
First we compute the strong Connes spectrum of the action
$\alpha^{\omega}:G\curvearrowright{\mathcal O}_\infty$.
We need the following lemma.
\begin{lemma}\label{H_csg}
For any $\omega\in{\mathcal G}amma^\infty$, we have
$\{0\}\cup H_{\overline{\rm{sg}}(\omega)}
=\{0\}\cup\bigcap_{n=1}^\infty\overline{\bigcup_{i=n}^\infty({\rm{sg}}(\omega)+\omega_i)}$.
\varepsilonnd{lemma}
\begin{proposition}f
It suffices to show that
$$\overline{\rm{sg}}(\omega)\setminus\big(\{0\}\cup\bigcup_{i=1}^\infty(\overline{\rm{sg}}(\omega)+\omega_i)\big)\subset
\overline{\bigcup_{i=n+1}^\infty({\rm{sg}}(\omega)+\omega_i)}$$
for any $n\in\mathbb{Z}_+$.
Take $\gamma\in\overline{\rm{sg}}(\omega)\setminus
\big(\{0\}\cup\bigcup_{i=1}^\infty(\overline{\rm{sg}}(\omega)+\omega_i)\big)$
and $n\in\mathbb{Z}_+$.
Since $\gamma\in\overline{\rm{sg}}(\omega)$, there exists a sequence $\{\mu_k\}\subset{\mathcal W}_\infty$
such that $\gamma=\lim_{k\to\infty}\omega_{\mu_k}$.
We will show that we can find an integer grater than $n$ in the word $\mu_k$
for infinitely many $k$, from which it follows that
$\gamma\in\overline{\bigcup_{i=n+1}^\infty({\rm{sg}}(\omega)+\omega_i)}$.
To the contrary, assume that $\mu_k\in{\mathcal W}_n$ for sufficiently large $k$.
Then there exists $i\in\{1,2,\ldots,n\}$ which appears in $\mu_k$ eventually.
We have $\gamma-\omega_i=\lim_{k\to\infty}(\omega_{\mu_k}-\omega_i)\in\overline{\rm{sg}}(\omega)$.
This contradicts the fact that $\gamma\notin\overline{\rm{sg}}(\omega)+\omega_i$.
Hence
$\{0\}\cup H_{\overline{\rm{sg}}(\omega)}
=\{0\}\cup\bigcap_{n=1}^\infty\overline{\bigcup_{i=n}^\infty({\rm{sg}}(\omega)+\omega_i)}$.
\varepsilonnd{proposition}f
\begin{proposition}\label{SCS}
The strong Connes spectrum $\widetilde{{\mathcal G}amma}(\alpha^{\omega})$
of the action $\alpha^{\omega}$ is $\{0\}\cup H_{\overline{\rm{sg}}(\omega)}$.
\varepsilonnd{proposition}
\begin{proposition}f
By \cite[Lemma 3.4]{Ki}, we have
$$\widetilde{{\mathcal G}amma}(\alpha^{\omega})
=\{\gamma\in{\mathcal G}amma\mid \widehat{\alpha^{\omega}}_\gamma(I)\subset I,
\mbox{for any ideal } I \mbox{ of } {\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G\},$$
where $\widehat{\alpha^{\omega}}:{\mathcal G}amma\curvearrowright{\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ is the dual
action of $\alpha^{\omega}$.
For an $\omega$-invariant pair $\widetilde{X}=(X,X^{(\infty)})$
and $\gamma\in{\mathcal G}amma$,
we see that $\widehat{\alpha^{\omega}}_\gamma(I_{\widetilde{X}})
=I_{\widetilde{X}-\gamma}$ where
$\widetilde{X}-\gamma=(X-\gamma,X^{(\infty)}-\gamma)$.
Hence $\widehat{\alpha^{\omega}}_\gamma(I_{\widetilde{X}})\subset
I_{\widetilde{X}}$ is equivalent
to say that $X+\gamma\subset X$ and $X^{(\infty)}+\gamma\subset X^{(\infty)}$
for an $\omega$-invariant pair $\widetilde{X}=(X,X^{(\infty)})$
and $\gamma\in{\mathcal G}amma$.
Considering the case that $\widetilde{X}=(\overline{\rm{sg}}(\omega),\{0\}\cup H_{\overline{\rm{sg}}(\omega)})$,
we have $(\{0\}\cup H_{\overline{\rm{sg}}(\omega)})+\gamma\subset\{0\}\cup H_{\overline{\rm{sg}}(\omega)}$
for $\gamma\in\widetilde{{\mathcal G}amma}(\alpha^{\omega})$.
Hence $\widetilde{{\mathcal G}amma}(\alpha^{\omega})\subset\{0\}\cup H_{\overline{\rm{sg}}(\omega)}$.
Let $(X,X^{(\infty)})$ be an $\omega$-invariant pair.
For $\gamma\in X$, we get
$$\gamma+\bigcap_{n=1}^\infty\overline{\bigcup_{i=n}^\infty({\rm{sg}}(\omega)+\omega_i)}
=\bigcap_{n=1}^\infty\overline{\bigcup_{i=n}^\infty(\gamma+{\rm{sg}}(\omega)+\omega_i)}
\subset\bigcap_{n=1}^\infty\overline{\bigcup_{i=n}^\infty(X+\omega_i)}
\subset H_X\subset X^{(\infty)}.$$
By Lemma \ref{H_csg}, we have
$X^{(\infty)}+(\{0\}\cup H_{\overline{\rm{sg}}(\omega)})\subset X^{(\infty)}$.
Since $\{0\}\cup H_{\overline{\rm{sg}}(\omega)}\subset\overline{\rm{sg}}(\omega)$,
we have $X+(\{0\}\cup H_{\overline{\rm{sg}}(\omega)})\subset X$.
Hence when $\omega$ satisfies Condition \ref{cond}, we have
$\widetilde{{\mathcal G}amma}(\alpha^{\omega})\supset \{0\}\cup H_{\overline{\rm{sg}}(\omega)}$
by Theorem \ref{idestr1},
and so $\widetilde{{\mathcal G}amma}(\alpha^{\omega})=\{0\}\cup H_{\overline{\rm{sg}}(\omega)}$.
Next we consider the case
that $\omega$ does not satisfy Condition \ref{cond}.
For an $\omega$-invariant pair $(Y,X^{(\infty)})$,
we have $X+(H_{\overline{\rm{sg}}(\omega)}\setminus\{0\})\subset X^{(\infty)}$
by the former part of this proof, where
$X=\{\gamma\in{\mathcal G}amma\mid
([\gamma],\theta)\in Y \mbox{ for some }\theta\in\mathbb{T}\}$.
Hence for any $([\gamma_0],\theta_0)\in Y$
and $\gamma\in H_{\overline{\rm{sg}}(\omega)}\setminus\{0\}$,
we have $\gamma_0+\gamma\in X^{(\infty)}$
because $\gamma_0\in X$.
Since $[X^{(\infty)}]\times\mathbb{T}\subset Y$, we have
$([\gamma_0+\gamma],\theta_0)\in Y$.
Therefore we also have
$\{0\}\cup H_{\overline{\rm{sg}}(\omega)}\subset\widetilde{{\mathcal G}amma}(\alpha^{\omega})$
by Theorem \ref{idestr2}.
Thus $\widetilde{{\mathcal G}amma}(\alpha^{\omega})=\{0\}\cup H_{\overline{\rm{sg}}(\omega)}$.
\varepsilonnd{proposition}f
Next we give necessary and sufficient conditions for $\omega\in{\mathcal G}amma^\infty$
that the crossed product ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ becomes simple or primitive.
\begin{lemma}\label{0}
Let $I$ be an ideal of the crossed product ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$.
Then $I=0$ if and only if $X_I={\mathcal G}amma$.
\varepsilonnd{lemma}
\begin{proposition}f
The ``only if'' part is trivial.
One can easily prove the ``if'' part
by the same arguments as in the proofs of
Proposition \ref{cond.exp2} and Theorem \ref{idestr1}.
\varepsilonnd{proposition}f
\begin{proposition}
For $\omega\in{\mathcal G}amma^\infty$, the following are equivalent:
\benu
\item The crossed product ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ is simple.
\item There are no $\omega$-invariants sets
other than ${\mathcal G}amma$ and $\varepsilonmptyset$.
\item ${\mathcal G}amma=\overline{\rm{sg}}(\omega)$.
\varepsilonnd{enumerate}
If ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ is simple, then it is purely infinite.
\varepsilonnd{proposition}
\begin{proposition}f
The equivalence between (i) and (ii) follows from Lemma \ref{0}.
(ii) implies (iii) because $\overline{\rm{sg}}(\omega)$ is $\omega$-invariant.
(iii) implies (ii) because $X=X+\overline{\rm{sg}}(\omega)$ if $X$ is $\omega$-invariant.
For the last statement, see \cite[Proposition 5.2]{Ka2}.
\varepsilonnd{proposition}f
The equivalence between (i) and (iii) was already proved
by A. Kishimoto \cite{Ki} by using strong Connes spectrum.
Note that the strong Connes spectrum $\widetilde{{\mathcal G}amma}(\alpha^{\omega})$
is equal to ${\mathcal G}amma$ if and only if $\overline{\rm{sg}}(\omega)={\mathcal G}amma$ by Proposition \ref{SCS}.
\begin{proposition}
The following conditions for $\omega\in{\mathcal G}amma^\infty$ are equivalent:
\benu
\item The crossed product ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ is primitive.
\item ${\mathcal G}amma$ is a prime $\omega$-invariant set.
\item The closed group generated by $\omega_1,\omega_2,\ldots$
is equal to ${\mathcal G}amma$.
\varepsilonnd{enumerate}
\varepsilonnd{proposition}
\begin{proposition}f
(i)$\mathbb{R}ightarrow$(ii): This follows from Proposition \ref{prime}.
(ii)$\mathbb{R}ightarrow$(i): It suffices to show that $0$ is prime.
Let $I_1,I_2$ be ideals of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ with $I_1\cap I_2=0$.
We have $X_{I_1}\cup X_{I_2}=X_{I_1\cap I_2}={\mathcal G}amma$.
Since ${\mathcal G}amma$ is prime, either $X_{I_1}\supset{\mathcal G}amma$ or $X_{I_2}\supset{\mathcal G}amma$.
If $X_{I_1}\supset{\mathcal G}amma$ hence $X_{I_1}={\mathcal G}amma$, then $I_1=0$ by Lemma \ref{0}.
Similarly if $X_{I_2}\supset{\mathcal G}amma$, then $I_2=0$.
Thus $0$ is prime and so ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ is a primitive $C^*$-algebra.
(ii)$\iff$(iii): This follows from Proposition \ref{Xprime}.
\varepsilonnd{proposition}f
One can prove the equivalence between (i) and (iii) in the above theorem
by characterization of primitivity of crossed products in terms of
the Connes spectrum due to D. Olesen and G. K. Pedersen \cite{OP} and
the computation of the Connes spectrum of our actions $\alpha^{\omega}$
due to A. Kishimoto \cite{Ki}.
\begin{proposition}\label{CP}
The crossed product ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ is isomorphic to
the Cuntz-Pimsner algebra ${\mathcal O}_E$ of $C_0({\mathcal G}amma)$-bimodule
$E=C_0({\mathcal G}amma)^\infty$, whose left module structure is given by
$$f\cdot(f_1,f_2,\ldots,f_n,\ldots)=(\sigma_{\omega_1}(f)f_1,\sigma_{\omega_2}(f)f_2,\ldots,\sigma_{\omega_n}(f)f_n,\ldots)\in E$$
for $f\in C_0({\mathcal G}amma)$ and $(f_1,f_2,\ldots,f_n,\ldots)\in E$.
\varepsilonnd{proposition}
\begin{proposition}f
The inclusion $C_0({\mathcal G}amma)\hookrightarrow{\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ and
$E\ni (0,\ldots,0,f_n,0\ldots)\mapsto S_nf_n\in{\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$
satisfies the conditions in \cite[Theorem 3.12]{Pi}.
Hence there exists a $*$-homomorphism
$\varphi:{\mathcal O}_E\to{\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ which is surjective
since ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ is generated by $\{S_nf\mid n\in\mathbb{Z}_+,\ f\in C_0({\mathcal G}amma)\}$.
One can show that $\varphi$ is injective by using Lemma \ref{isom}.
Thus ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ is isomorphic to ${\mathcal O}_E$.
\varepsilonnd{proposition}f
\begin{corollary}
The inclusion $C_0({\mathcal G}amma)\hookrightarrow{\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ is a KK-equivalence.
Hence for $i=0,1$, we have $K_i({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)=K_i(C_0({\mathcal G}amma))$.
\varepsilonnd{corollary}
\begin{proposition}f
See \cite[Corollary 4.5]{Pi}.
\varepsilonnd{proposition}f
\begin{proposition}\label{embedinfty}
If $\omega\in{\mathcal G}amma^\infty$ satisfies
$-\omega_i\notin\overline{\{\omega_{\mu}\mid\mu\in{\mathcal W}_n\}}$
for any $i,n\in\mathbb{Z}_+$,
then the crossed product ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ is AF-embeddable.
\varepsilonnd{proposition}
\begin{proposition}f
See \cite[Proposition 5.1]{Ka2}.
\varepsilonnd{proposition}f
\begin{thebibliography}{BHRS}
\bibitem[BHRS]{BHRS}
Bates, T.; Hong, J.; Raeburn, I.; Szyma\'nski, W. {\it The ideal structure of the C*-algebras of infinite graphs.} Preprint.
\bibitem[C]{C}
Cuntz, J. {\it Simple $C\sp*$-algebras generated by isometries.} Comm. Math. Phys. {\bf 57} (1977), no. 2, 173--185.
\bibitem[D]{D}
Deaconu, V. {\it Continuous graphs and C*-algebras.} Operator theoretical methods, 137--149, Theta Found., Bucharest, 2000.
\bibitem[E]{E}
Evans, D. E. {\it On $O\sb{n}$.} Publ. Res. Inst. Math. Sci. {\bf 16} (1980), no. 3, 915--927.
\bibitem[HS]{HS}
Hong, J. H.; Szymanski, W. {\it The primitive ideal space of the $C^*$-algebras of infinite graphs.} Preprint.
\bibitem[Ka1]{Ka1}
Katsura, T. {\it The ideal structures of crossed products of Cuntz algebras by quasi-free actions of abelian groups.} Preprint.
\bibitem[Ka2]{Ka2}
Katsura, T. {\it AF-embeddability of crossed products of Cuntz algebras.} Preprint.
| 4,022 | 50,987 |
en
|
train
|
0.36.14
|
(ii)$\mathbb{R}ightarrow$(i): It suffices to show that $0$ is prime.
Let $I_1,I_2$ be ideals of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ with $I_1\cap I_2=0$.
We have $X_{I_1}\cup X_{I_2}=X_{I_1\cap I_2}={\mathcal G}amma$.
Since ${\mathcal G}amma$ is prime, either $X_{I_1}\supset{\mathcal G}amma$ or $X_{I_2}\supset{\mathcal G}amma$.
If $X_{I_1}\supset{\mathcal G}amma$ hence $X_{I_1}={\mathcal G}amma$, then $I_1=0$ by Lemma \ref{0}.
Similarly if $X_{I_2}\supset{\mathcal G}amma$, then $I_2=0$.
Thus $0$ is prime and so ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ is a primitive $C^*$-algebra.
(ii)$\iff$(iii): This follows from Proposition \ref{Xprime}.
\varepsilonnd{proposition}f
One can prove the equivalence between (i) and (iii) in the above theorem
by characterization of primitivity of crossed products in terms of
the Connes spectrum due to D. Olesen and G. K. Pedersen \cite{OP} and
the computation of the Connes spectrum of our actions $\alpha^{\omega}$
due to A. Kishimoto \cite{Ki}.
\begin{proposition}\label{CP}
The crossed product ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ is isomorphic to
the Cuntz-Pimsner algebra ${\mathcal O}_E$ of $C_0({\mathcal G}amma)$-bimodule
$E=C_0({\mathcal G}amma)^\infty$, whose left module structure is given by
$$f\cdot(f_1,f_2,\ldots,f_n,\ldots)=(\sigma_{\omega_1}(f)f_1,\sigma_{\omega_2}(f)f_2,\ldots,\sigma_{\omega_n}(f)f_n,\ldots)\in E$$
for $f\in C_0({\mathcal G}amma)$ and $(f_1,f_2,\ldots,f_n,\ldots)\in E$.
\varepsilonnd{proposition}
\begin{proposition}f
The inclusion $C_0({\mathcal G}amma)\hookrightarrow{\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ and
$E\ni (0,\ldots,0,f_n,0\ldots)\mapsto S_nf_n\in{\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$
satisfies the conditions in \cite[Theorem 3.12]{Pi}.
Hence there exists a $*$-homomorphism
$\varphi:{\mathcal O}_E\to{\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ which is surjective
since ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ is generated by $\{S_nf\mid n\in\mathbb{Z}_+,\ f\in C_0({\mathcal G}amma)\}$.
One can show that $\varphi$ is injective by using Lemma \ref{isom}.
Thus ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ is isomorphic to ${\mathcal O}_E$.
\varepsilonnd{proposition}f
\begin{corollary}
The inclusion $C_0({\mathcal G}amma)\hookrightarrow{\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ is a KK-equivalence.
Hence for $i=0,1$, we have $K_i({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)=K_i(C_0({\mathcal G}amma))$.
\varepsilonnd{corollary}
\begin{proposition}f
See \cite[Corollary 4.5]{Pi}.
\varepsilonnd{proposition}f
\begin{proposition}\label{embedinfty}
If $\omega\in{\mathcal G}amma^\infty$ satisfies
$-\omega_i\notin\overline{\{\omega_{\mu}\mid\mu\in{\mathcal W}_n\}}$
for any $i,n\in\mathbb{Z}_+$,
then the crossed product ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ is AF-embeddable.
\varepsilonnd{proposition}
\begin{proposition}f
See \cite[Proposition 5.1]{Ka2}.
\varepsilonnd{proposition}f
\begin{thebibliography}{BHRS}
\bibitem[BHRS]{BHRS}
Bates, T.; Hong, J.; Raeburn, I.; Szyma\'nski, W. {\it The ideal structure of the C*-algebras of infinite graphs.} Preprint.
\bibitem[C]{C}
Cuntz, J. {\it Simple $C\sp*$-algebras generated by isometries.} Comm. Math. Phys. {\bf 57} (1977), no. 2, 173--185.
\bibitem[D]{D}
Deaconu, V. {\it Continuous graphs and C*-algebras.} Operator theoretical methods, 137--149, Theta Found., Bucharest, 2000.
\bibitem[E]{E}
Evans, D. E. {\it On $O\sb{n}$.} Publ. Res. Inst. Math. Sci. {\bf 16} (1980), no. 3, 915--927.
\bibitem[HS]{HS}
Hong, J. H.; Szymanski, W. {\it The primitive ideal space of the $C^*$-algebras of infinite graphs.} Preprint.
\bibitem[Ka1]{Ka1}
Katsura, T. {\it The ideal structures of crossed products of Cuntz algebras by quasi-free actions of abelian groups.} Preprint.
\bibitem[Ka2]{Ka2}
Katsura, T. {\it AF-embeddability of crossed products of Cuntz algebras.} Preprint.
\bibitem[Ki]{Ki}
Kishimoto, A. {\it Simple crossed products of $C\sp{*} $-algebras by locally compact abelian groups.} Yokohama Math. J. {\bf 28} (1980), no. 1-2, 69--85.
\bibitem[KK1]{KK1}
Kishimoto, A.; Kumjian, A. {\it Simple stably projectionless $C\sp *$-algebras arising as crossed products.} Canad. J. Math. {\bf 48} (1996), no. 5, 980--996.
\bibitem[KK2]{KK2}
Kishimoto, A.; Kumjian, A. {\it Crossed products of Cuntz algebras by quasi-free automorphisms.} Operator algebras and their applications, 173--192, Fields Inst. Commun., {\bf 13}, Amer. Math. Soc., Providence, RI, 1997.
\bibitem[OP]{OP}
Olesen, D.; Pedersen, G. K. {\it Applications of the Connes spectrum to $C\sp{*} $-dynamical systems.} J. Funct. Anal. {\bf 30} (1978), no. 2, 179--197.
\bibitem[Pi]{Pi}
Pimsner, M. V. {\it A class of $C\sp *$-algebras generalizing both Cuntz-Krieger algebras and crossed products by ${Z}$.} Free probability theory, 189--212, Fields Inst. Commun., {\bf 12}, Amer. Math. Soc., Providence, RI, 1997.
\varepsilonnd{thebibliography}
\varepsilonnd{document}
| 1,960 | 50,987 |
en
|
train
|
0.37.0
|
\begin{document}
\title{A toy model for Macroscopic Quantum Coherence}
\author{R. Mu\~{n}oz-Vega}\email{[email protected]}
\affiliation{Universidad Aut\'{o}noma de la Ciudad de M\'{e}xico,
Centro Hist\'{o}rico, Fray Servando Teresa de Mier 92,
Col. Centro, Del. Cuauht\'{e}moc, M\'{e}xico D.F, C.P. 06080}
\author{Jos\'{e} Job Flores-Godoy}\email{ e-mail:[email protected]}
\author{ G. Fern\'{a}ndez-Anaya}\email{[email protected]}
\affiliation{Departamento de F\'isica y Matem\'aticas, Universidad Iberoamericana, Prol. Paseo de la Reforma 880, Col. Lomas de Santa Fe, Del. A. Obreg\'on, M\'exico, D. F. C.P. 01219}
\author{Encarnaci\'{o}n Salinas-Hern\'{a}ndez}
\email{ [email protected]}
\affiliation{ESCOM-IPN,
Av. Juan de Dios B\'{a}tiz s/n, Unidad Profesional Adolfo L\'{o}pez Mateos
Col. Lindavista, Del. G. A. Madero, M\'{e}xico, D. F, C.P. 07738}
\begin{abstract}
The present article deals with Macroscopic Quantum Coherence resorting only to basic quantum mechanics. A square double well is used to illustrate the Leggett-Caldeira oscillations. The effect of thermal-radiation on two-level systems is discussed to some extent. The concept of decoherence is introduced at an elementary level. Handles are deduced for the energy, temperature and time scales involved in Macroscopic Quantum Coherence.
\end{abstract}
\date{\today}
\pacs{01.40.Ha, 03.67.-a, 03.65.Fd, 03.65.Ge, 02.10.Ud, 03.65.Ca}
\maketitle
\section{Introduction}
Triggered by a seminal article \cite{Leggett} written by A J Leggett in 1980, research into Macroscopic Quantum Coherence (MQC) has yielded impressive experimental,\cite{Nakamuraetal, Makhlinetal, Friedmanetal, WS} theoretical\cite{CaldeiraLeggett, Leggett1, RevDiss, Tesche} and even technological achievements\cite{Carellietal, Manucharyanetal01, Manucharyanetal02}. The ideas developed in the last thirty so years by Leggett and his collaborators have not only changed the way we understand the relation between quantum and classical behaviours, but are also crucial in the future development of quantum computing. The present article aims at explaining the basic phenomenology of QMC resorting only to basic quantum mechanics. Thus, we believe this article can be of interest for any student who has attended at least a one-year course in quantum physics, and for faculty members committed to introducing students into contemporary research.
In order to explain briefly what MQC is, let us consider a particle in a symmetric double well potential (SDWP). Figure 1 depicts an example of such a potential. In freshmen courses we have been told what to expect when the particle is in a high-lying energy level in a nice, analytical, potential such as this: for states for which the change in potential energy within a de Broglie wavelength is much smaller than the mean kinetic energy, the specifically quantum features of the behavior result negligible and the classical description becomes adequate.\cite{Bohm} In that sense, classical behaviour can be considered as a limiting case of quantum mechanics \cite{LandauLif}. Suppose, nonetheless, the central barrier in the SDWP of Fig. 1 to be of macroscopic width. Then, the predictions of quantum mechanics and classical mechanics certainly clash for this system. A classical viewpoint would demand two distinct localized states of stable equilibrium, situated at $-x_{0}$ and $x_{0}$, while quantum mechanics predicts an even probability distribution for the (non-degenerate\cite{LandauA}) ground level state (which is, of course, the more stable stationary state.) In fact, the ground statefunction for such a potential would have to look something like Figure 2.
Moreover, the (odd) eigenfunction of the first excited level (Figure 3), and indeed each one of the stationary solutions of a SDWP, necessarily has an even probability distribution.
\begin{figure}
\caption{An example of a SDWP potential, with characteristic double minima and central peak.}
\label{fig:figure1}
\end{figure}
What Leggett predicted more than thirty wears ago, and what actually happens in experiments carried out in SDWPs of micrometric and nanometric typical lengths, is the appearance of a two-fold degenerate ground level $E^{\prime}$, with the system oscillating in an harmonic fashion between two eigenstates, $\vert L\rangle$ (Figure 4) and $\vert R \rangle$ (Figure 5) localized, respectively, at the left and right of the central barrier. At ground level, the position expectancy value oscillates in the accordance with:
\begin{equation}\label{I.1}
\langle x\rangle_{0}(t)=\langle x\rangle_{0}(0)\cos \omega t\textrm{.}
\end{equation}
This phenomenon, the so called Leggett-Caldeira oscillations, is closely related with the Rabi oscillations of atomic physics. It is explained as the result of the purported ground level $E^{\prime}$ resolving into a true ground level
\begin{equation}\label{I.2}
E_{+}=E^{\prime}-\hbar \omega/2\textrm{,}
\end{equation}
endowed with an even non-localized eigensolution $\vert +\rangle$, and a first excited level
\begin{equation}\label{I.3}
E_{-}=E^{\prime}+\hbar \omega/2\textrm{,}
\end{equation}
endowed with an odd non-localized eigensolution $\vert -\rangle$. When a quantum system tunnels periodically trough the barrier of a SDWP with a central barrier of macroscopic length, we have Macroscopic Quantum Coherence.
The states $\vert R\rangle$ and $\vert L\rangle$ have, each one on its own, a definite value of a macroscopic property (namely, the property of being localized at the left or the right of the barrier). At the same time, $\vert R \rangle$ and $\vert L\rangle$ are linear combinations of the states $\vert +\rangle$ and $\vert -\rangle$, which cannot be said to be localized. In order to understand Leggett's original motivation, notice the analogy between macroscopic SDWPs and Schroedinger's cat: the celebrated pet can be in any of two different \emph{macroscopically distinguishable} states (let us say, $\Psi_{1}$ for a live cat and $\Psi_{0}$ for a dead one) just as a particle in a SDWP. If any of these macroscopic systems obeys the laws of quantum mechanics, then it could be prepared in linear combinations that lack a sharp, well defined, value of the macroscopic property. Examples of these linear combinations are the $\vert\pm\rangle$ states of the SDWPs, and the ``neither dead nor alive" states
\begin{equation}
\Psi_{\pm}=\frac{1}{\sqrt{2}}\Big(\Psi_{0}\pm\Psi_{1}\Big)
\end{equation}
of the cat.
\begin{figure}
\caption{A rendering of what a ground-level eigenfunction (solid curve) should look like for a SDWP. The potential is shown as a dashed curve.}
\label{fig:B}
\label{fig:figure1}
\end{figure}
Thus, a more general definition of MQC is simply: the quantum superposition of distinct macroscopic states. Long time before the year of 1980, macroscopic quantum phenomena had been discovered: superconductivity in 1911, and superfluidity in 1937. Yet it remained for Leggett to identify the conditions necessary for a quantum system to present macroscopically distinguishable states.\cite{Leggett}
Some twenty years elapsed between Leggett's proposal and a credible experimental confirmation\cite{Friedmanetal, WS} of MQC. One of the main reasons for this delay lies in the fact that the phase coherence of the $\vert\pm\rangle$ states is rapidly lost due to the interaction of the system with its surroundings, so the system collapses into one of the localized states before one period of the Leggett-Caldeira oscillation is completed.\cite{Leggett, RevDiss}
MQC is relevant not only from the purely theoretical point of view. A physical qubit is a two-level system considered as a piece of hardware. And, as we shall see in the following pages, at least some SDWPs can behave as effective two-level systems at sufficiently low temperatures. Quantum computing (an area with impressive software development, but little hardware to show) requires qubits to interact with one another without loss of coherence, for fairly long times, even at fairly high temperatures. Thus, the study of two-level dissipative systems, to which Leggett and collaborators made far reaching contributions when delving in the foundations of quantum physics, has revealed itself crucial for people in the vanguard of technological development.\cite{Makhlinetal, WS}
\begin{figure}
\caption{Th first excited level eigenfunction (solid) of a SDWP (dashed).}
\label{fig:C}
\end{figure}
The rest of this article is structured as follows: in section~\ref{sec:2} we discuss the spectra of a family of symmetric double square well potentials, and the conditions under which a member of this family can be considered as an effective two-level system. Next, the properties of two-states systems arising from SDWP´s are discussed in section ~\ref{sec:3}. We then go on to examine in ~\ref{sec:4} how thermal radiation, by throwing the system into higher energy levels, renders the two-level model inapplicable. In section ~\ref{sec:5} decoherence is introduced in elementary terms, and its relation with dissipation is briefly discussed. Handles for the time, energy and temperature scales involved in MQC are derived from our toy model in section ~\ref{sec:6}. Finally, conclusions are laid down in section ~\ref{sec:7}.
\begin{figure}
\caption{The $\langle x\vert L\rangle$ state (shown solid) localized at the left of the SDWP (shown dashed) barrier.}
\label{fig:D}
\end{figure}
\begin{figure}
\caption{The $\langle x\vert R\rangle$ state (shown solid) localized at the right of the SDWP (shown dashed) barrier.}
\label{fig:E}
\end{figure}
| 2,719 | 17,658 |
en
|
train
|
0.37.1
|
\section{Symmetric Double Square Wells}\label{sec:2}
Leggett resorted to quasi-classical considerations when stating his original proposal.\cite{Leggett} Also, the WKB approximation has been applied to double well potentials by Landau and Lifshitz,\cite{LLandau} and more recently, in this Journal, by others.\cite{JelicMarsiglio} Here will take a different point of view, avoiding all together quasi-classical approximations, by considering a particular family of double infinite square well potentials as approximations to actual, analytic SDWPs. Our procedure will later allow us to get some reference values on the energies, temperatures and times involved in MQC. The following family of piece-wise-constant potentials will be considered:
\begin{equation}\label{ucases}
U_{b}(x)=\left \{ \begin{array}{c l}
\infty &\textrm{if } x \leq -a-b\textrm{,}\\
0 &\textrm{if } -b > x > -a-b\textrm{,}\\
k &\textrm{if } b\geq x \geq -b\textrm{,}\\
0 &\textrm{if } b+a> x > b\textrm{,}\\
\infty &\textrm{if } x \geq b+a\textrm{.}\\
\end{array}\right .
\end{equation}
Potentials of this kind have previously been studied in a different context, and it has been shown\cite{Munoz} that, if all other parameters held fixed, levels $E_{2n+1}$ and $E_{2n}$ coalesce as $k\rightarrow \infty$. Here we shall consider the barrier height $k>0$ as a fixed number, although ``big" in a sense that will be readily clarified. This, in order to keep the gap between the ground and first excited levels sufficiently small. We shall also take the width of each one of the lateral valleys, $a>0$, as a fixed value unless otherwise stated, leaving free the only other parameter, that is the barrier half-width $b>0$.
One of the two main objectives of this Section is to obtain a global lower bound on the energy gap between the first and second excited levels in the $U_{b}$ potentials. Just as important to our ends, we will learn on this Section that there is a ``running" upper bound (dependent on the value of $b$) on the gap between the ground and first excited levels. The consequences of this to facts, which are vital to the rest of the article, are explored in Sections III, IV and VI.
To be sure, non of the $U_{b}$ is continuous, yet they share the most prominent features of a SDWP, namely, they are even potentials with completely bounded, non-degenerate, spectra, as can be shown from boundary conditions. If instead of two minima, the $U_{b}$ have two non-overlapping regions of minima (\emph{viz.} $(-a-b,-b)$ and $(b,a+b)$), this distinction will prove to be quite unimportant.
\begin{figure}
\caption{A typical member of the $U_b(x)$ family of potentials}
\label{msbs1}
\label{fig:figure5r}
\end{figure}
Also from boundary conditions (or from more abstract, symmetry considerations) it is readily seen that the levels in the spectrum of any of the $U_{b}$ are classified according with parity, just like it happens for a continuous even potential:
\begin{equation}\label{II.1}
\psi_{2n,b}(-x)=\psi_{2n,b}(x)\textrm{,}\quad\quad n=0,1,\ldots\ \textrm{,}\quad b\in (0, \infty )
\end{equation}
and
\begin{equation}\label{II.2}
\psi_{2n+1,b}(-x)=-\psi_{2n+1,b}(x)\textrm{,}\quad\quad n=0,1,\ldots\ \textrm{,}\quad b\in (0, \infty )\textrm{.}
\end{equation}
Let us focus on the discretization conditions below the level of the central barrier ($E<k$). From the boundary conditions we get, for even states:
\begin{equation}\label{coneven}
-\sqrt{E_{2n}}\cot a\frac{\sqrt{2mE_{2n}}}{\hbar}=\sqrt{k-E_{2n}}\tanh b\frac{\sqrt{2m(k-E_{2n})}}{\hbar}\quad\textrm{,}
\end{equation}
while odd levels below the barrier level have to comply with
\begin{equation}\label{conodd}
-\sqrt{E_{2n+1}}\cot a\frac{\sqrt{2mE_{2n+1}}}{\hbar}=\sqrt{k-E_{2n+1}}\coth b\frac{\sqrt{2m(k-E_{2n+1})}}{\hbar}\quad\textrm{.}
\end{equation}
Notice how the first of these two conditions can be written in the form:
\begin{equation}
g(E_{2n})=h_{b}(E_{2n}),
\end{equation}
and the second can be rendered as:
\begin{equation}
g(E_{2n+1})=j_{b}(E_{2n+1}),
\end{equation}
with the meaning of $g, h_{b}$ and $j_{b}$ being obvious from the context.
Both of these two last equations are depicted in Figure 7, from which it can be seen that there exists an upper bound $B$, given by
\begin{equation}\label{inequalityA}
B=\frac{\pi^{2}\hbar ^{2}}{2m a^{2}},
\end{equation}
such that the ground and first excited states have to comply with
\begin{equation}\label{funnyeq}
\frac{B}{4}<E_{0}<E_{1}<B ,
\end{equation}
no matter the value of $b.$ Obviously, there can be no levels below the barrier unless $k>B/4$. We shall only consider potentials for which the condition:
\begin{equation}\label{bigenough}
k>>B
\end{equation}
is met, so that we will always have at least two levels below the barrier. Indeed, the number of levels below the barrier increases with increasing quotient $k/B$ and, more importantly, as $B$ is independent of $k$, condition (\ref{bigenough}) warrants that the gap between the first two level is always small.
It is not difficult to generalize (\ref{funnyeq}) starting from (\ref{coneven}) and (\ref{conodd}) and definition (\ref{inequalityA}). The result is that:
\begin{equation}\label{III.1.1}
(n+\frac{1}{2})^{2}B<E_{2n,k}<E_{2n+1,k}<(n+1)^{2}B\textrm{,}\quad n=0,1,2,\ldots N,
\end{equation}
if the level $2N+1$ is still below the barrier.
From inequality (\ref{III.1.1}) it follows that
\begin{equation}\label{III.1.2}
E_{2n+2}-E_{2n+1}>(n+5/4)B\textrm{,}\quad n=0,1,2,\ldots, N
\end{equation}
if level $2N+1$ is below the barrier. We then have that the gap between the ground and first excited levels will always be less than the gap between the first and second excited levels:
\begin{equation}\label{inek01}
E_{2}-E_{1}>\frac{5}{4}B>\frac{3}{4}B> E_{1}-E_{0}.
\end{equation}
But we can do much more better than that. Indeed, in Appendix A it is formally proven that for any given number $\delta>0$ there exist a value $b(\delta)>0$ such the gap between the ground and the first excited level of a $U_{b}$ potential will be less than $\delta$, that is
\begin{equation}
E_{1}-E_{0}<\delta ,
\end{equation}
if $b\geq b(\delta)$. In other words, if we choose the barrier length big enough, then we can make $E_{0}$ and $E_{1}$ as proximate as we want, while there is a lower bound for the gap between $E_{2}$ and $E_{1}$ which is independent of the value of this length. This will allow us to find examples of $U_{b}$ that will work as effective two-state systems for the lowest-lying energy levels, as illustrated in Figure 8.
\begin{figure}
\caption{Graphical solutions of transcendental equations (\ref{coneven}
\end{figure}
\begin{figure}
\caption{Doubling the barrier width produces a dramatic decrease in the gap between the ground and first excited levels. Shown: function $h_{b}
\end{figure}
Finally, there is one more inequality that can be derived from (\ref{III.1.1}) and that will prove useful in section IV. This inequality is:
\begin{equation}\label{2NF}
E_{2}-E_{1}<\frac{15}{4}B\ .
\end{equation}
Let us stress that relations (\ref{funnyeq}), (\ref{inek01}) and (\ref{2NF}) are verified for each $U_{b}$ regardless of the value of $b.$
\begin{figure}
\caption{Solid curve: the $\psi_{+}
\label{fig:figure1}
\end{figure}
| 2,301 | 17,658 |
en
|
train
|
0.37.2
|
\section{Two-level systems with reflection symmetry}\label{sec:3}
In the preceding section we have proven that there are $U_{b}$ potentials for which the gap between the ground and first excited energy levels is much more narrow than the one between the first and second excited levels. Consequently, for low energy expectancy values, a particle in one of such potentials acts as an effective two-level systems.\cite{Feynman, Cohen}
In the rest of this section we shall consider a fixed $U_{b}$ that behaves as a two-level system, and drop the $b$.
Consider now the non-stationary solutions $\psi_{L}$ and $\psi_{R}$ that one obtains from the linear combinations
\begin{equation}\label{II.3}
\psi_{L} (x,t)=\frac{1}{\sqrt{2}}\Big [\exp \left(-\imath\frac{E_{0}t}{\hbar}\right) \psi_{0}(x) + \exp\left(-\imath\frac{E_{1}t}{\hbar}\right)\psi_{1}(x)\Big ]
\end{equation}
\begin{figure}
\caption{ Solid: the (odd) eigenfunction of the first excited level. Dashed: the $U_{b}
\end{figure}
and
\begin{equation}\label{II.4}
\psi_{R} (x,t)=\frac{1}{\sqrt{2}}\Big [ \exp\left(-\imath\frac{E_{0}t}{\hbar}\right) \psi_{0}(x) - \exp\left(-\imath\frac{E_{1}t}{\hbar}\right) \psi_{1}(x)\Big]\textrm{.}
\end{equation}
These states have no definite parity, but instead one is the specular image of the other:
\begin{equation}\label{II.A.1}
\psi_{L}(-x,t)=\psi_{R}(x,t)\textrm{,}
\end{equation}
as can be seen from equations (\ref{II.1}), (\ref{II.2}), (\ref{II.3}) and (\ref{II.4}).
The position expectancy value for this states is calculated from (\ref{II.1}) in a straightforward manner:
\begin{equation}\label{II.5}
\langle x\rangle_{L}(t) =\ -\ \langle x\rangle_{R}(t) =\ \langle\psi_{0}\vert x\vert \psi_{1}\rangle\cos\frac{E_{1}-E_{0}}{\hbar}t,
\end{equation}
as is the energy expectancy value:
\begin{equation}\label{II.6}
\langle H\rangle_{L}=\langle H \rangle_{R}=\frac{E_{0}+E_{1}}{2}\textrm{.}
\end{equation}
Comparing (\ref{II.5}) with (\ref{I.1}) and (\ref{II.6}) with (\ref{I.2}) one may be tempted to make the identifications
\begin{equation}\label{II.7}
E^{\prime}=\langle H\rangle_{L}\quad\textrm{ and }\quad \omega=\frac{E_{1}-E_{2}}{\hbar}\textrm{,}
\end{equation}
from which (\ref{I.3}) would follow, so that the states of (\ref{II.3}) and (\ref{II.4}) could be interpreted as the localized states observed in the experiments, and $\psi_{0}$ and $\psi_{1}$ would correspond to the true ground level $E_{+}$ and the first excited state $E_{-}$. That is, it would be cogent that
\begin{equation}
\langle x\vert L\rangle=\psi_{L}(x)\textrm{,}\quad\langle x\vert R\rangle=\psi_{R}(x)\textrm{,}\quad
\langle x\vert +\rangle=\psi_{0}(x)\textrm{,}\quad\langle x\vert -\rangle=\psi_{1}(x)\textrm{.}
\end{equation}
\begin{figure}
\caption{The localized $\psi_{L}
\end{figure}
In this interpretation, however, there is no room for transitions. Indeed, the complete Schroedinger equation for a $U$ potential, which reads:
\begin{equation}\label{SchU}
-\frac{\hbar^{2}}{2m}\frac{\partial^{2}\psi}{\partial x^{2}}(x,t)+U(x)\psi(x,t)=i\hbar\frac{\partial\psi}{\partial t}(x,t) \textrm{,}
\end{equation}
predicts that if the system is initially prepared in the state $\psi_{L}(x,t=0)$ at time $t=0$, then it will remain in the $\psi_{L}(x,t)$ state for $t\in[0,\infty)$ (which is a sophisticated way to say: \emph{forever}). This is just consequence of PDE's theory.
Instead of periodical transitions between two different states, our equations predict the existence of a unique ``oscillating" state, because $\psi_{R}(x,t)$ is a time-displaced replica of $\psi_{L}(x,t)$:
\begin{figure}
\caption{The localized $\psi_{R}
\end{figure}
\begin{equation}\label{II.F}
\psi_{L}\Big(x,\ t+\frac{\pi}{\omega}\Big)=\imath\exp\Big(-\imath\frac{\pi\Omega}{\omega}\Big)\ \psi_{R}(x,t) \quad\textrm{ ,}
\end{equation}
where $\Omega$ stands for
\begin{equation}\label{II.G}
\Omega=(E_{1}+E_{2})/2\hbar
\end{equation}
and $\omega$ is as in (\ref{II.7}). One arrives at this result directly from (\ref{II.3}) and (\ref{II.4}) after some algebra.
\subsection{Flip-flops and Leggett-Caldeira oscillations}
Let us start from what we know happens in actual experiments (\emph{i. e.} the existence of an observable degenerate ground level) and proceed to deduce from there the perturbation needed to achieve such degeneracy. The SDWP Hamiltonian $H$ is represented by the matrix
\begin{equation}\label{IV.A.a}
\mathbb{H}=
\left(\begin{array}{cc}
E_{0}&0\\
0&E_{1}\\
\end{array}\right)
\end{equation}
in the symmetry-respecting basis formed by the eigenfunctions $\psi_{0}$ and $\psi_{1}$. Let us consider another Hamiltonian, $H^{\prime}$, represented by the matrix
\begin{equation}\label{IV.B.4}
\tilde{\mathbb{H}}^{\prime}=\mathbb{O}\mathbb{H}^{\prime}\mathbb{O}^{-1}=
\left(\begin{array}{cc}
E^{\prime}&0\\
0&E^{\prime}\\
\end{array}\right)
\end{equation}
in the symmetry-violating basis spanned by $\psi_{L}$ and $\psi_{R}$. Here, $\mathbb{O}$ stands for the unitary operator which transforms $\psi_{0}$ into $\psi_{L}$ and $\psi_{1}$ into $\psi_{R}$, thus:
\begin{equation}\label{IV.A.5}
\mathbb{O}\left(\begin{array}{c}
1\\
0\\
\end{array}\right)=\frac{1}{\sqrt{2}}\left(\begin{array}{c}
1\\
1\\
\end{array}\right)
\quad\textrm{ and }\quad
\mathbb{O}\left(\begin{array}{c}
0\\
1\\
\end{array}\right)=
\frac{1}{\sqrt{2}}\left(\begin{array}{c}
1\\
-1\\
\end{array}\right).
\end{equation}
Notice that, as the rhs of (\ref{IV.B.4}) is proportional to the identity matrix, then:
\begin{equation}
\mathbb{H}^{\prime}=\left(\begin{array}{cc}
E^{\prime}&0\\
0&E^{\prime}\\
\end{array}\right).
\end{equation}
Thus, the perturbation is represented by:
\begin{equation}
\mathbb{W}=\mathbb{H}^{\prime}-\mathbb{H}=\left(\begin{array}{cc}
\hbar \omega/2&0\\
0&-\hbar \omega/2\\
\end{array}\right).
\end{equation}
Now, in the basis spanned by $\psi_{L}$ and $\psi_{R}$ the things look quite different. Indeed, we have that:
\begin{equation}
\tilde{\mathbb{H}}=\mathbb{OHO}^{-1}=\left(\begin{array}{cc}
E^{\prime}&-\hbar \omega/2\\
-\hbar \omega/2&E^{\prime}\\
\end{array}\right)
\end{equation}
and most importantly:
\begin{equation}
\tilde{\mathbb{W}}=\mathbb{OWO}^{-1}=\left(\begin{array}{cc}
0&\hbar \omega/2\\
\hbar \omega/2&0\\
\end{array}\right).
\end{equation}
So that the perturbation has no diagonal elements. This means that zeroth order corrections are strictly null for the perturbation, and, further more, that the off-diagonal elements are equal.
To be very clear, let us write the eigen-equations for each one of this distinct systems. For $H$ we have
\begin{equation}\begin{array}{cc}
H\psi_{0}(x,t)=i\hbar\frac{\partial\psi_{0}}{\partial t}(x,t)=E_{0}\psi_{0}(x,t),&H\psi_{1}(x,t)=i\hbar\frac{\partial\psi_{1}}{\partial t}(x,t)=E_{1}\psi_{1}(x,t),\\
\end{array}
\end{equation}
while $H^{\prime}$ responds to
\begin{equation}\label{IV.Z}
\begin{array}{cc}
H^{\prime}\psi_{L}(x,t)=i\hbar\frac{\partial\psi_{L}}{\partial t}(x,t)=E^{\prime}\psi_{L}(x,t),&H^{\prime}\psi_{R}(x,t)=i\hbar\frac{\partial\psi_{R}}{\partial t}(x,t)=E^{\prime}\psi_{R}(x,t).\\
\end{array}
\end{equation}
Now, let us consider $\tilde{\mathbb{H}}^{\prime}$ as the initial, unperturbed, Hamiltonian matrix, and
\begin{equation}
-\tilde{\mathbb{W}}=-\mathbb{OWO}^{-1}
\end{equation}
as the perturbation, so that $\tilde{\mathbb{H}}$ is the final, perturbed, Hamiltonian matrix. Then we can show that the $\psi_{L}(x,t)$ and $\psi_{R}(x,t)$ states transit from one another in Rabi style. Indeed, resorting to the time-dependent perturbation formalism,\cite{Landau2, Fitzpatrick1} we write, for a general state $\psi(x,t)$ of $\tilde{\mathbb{H}},$
\begin{equation}\label{IV.A.7}
\psi(x,t)=c_{L}(t)\psi_{L}(x,t)+c_{R}(t)\psi_{R}(x,t),
\end{equation}
in order to obtain the equation
\begin{equation}\label{mtdpC}
i\hbar\frac{d}{dt}\left(\begin{array}{c}
c_{0}\\
c_{1}\\
\end{array}\right)(t)=\tilde{\mathbb{W}}\left(\begin{array}{c}
c_{0}\\
c_{1}\\
\end{array}\right)(t),
\end{equation}
which is equivalent to the $2\times 2$ system of coupled linear equations:
\begin{equation}\label{IV.A.8}
i\hbar\frac{dc_{L}}{dt}=-\frac{\hbar \omega}{2}c_{R}\quad , i\hbar\frac{dc_{R}}{dt}=-\frac{\hbar \omega}{2}c_{L}.\\
\end{equation}
By uncoupling this system we get the harmonic oscillator equation
\begin{equation}
\frac{d^{2}c_{L}}{dt^{2}}=-\frac{\omega^{2}}{4}c_{L}
\end{equation}
and a similar equation for $c_{R}$, so that
\begin{equation}\label{Rabires}
c_{L}(t)=\sin\big(\omega t/2+\phi\big)\quad c_{R}(t)=\cos\big(\omega t/2+\phi\big),
\end{equation}
where $\phi$ is a constant that can be elucidate from initial conditions. The probability of finding the particle in the state $\psi_{L}$ is given, according to this last equations, by
\begin{equation}\label{probell}
P_{L}(t)=\sin^{2}\big(\omega t/2+\phi\big),
\end{equation}
and the probability of finding the the particle in the $\psi_{R}$ state is
\begin{equation}\label{probar}
P_{R}(t)=1-P_{L}(t).
\end{equation}
This is a particular instance of the Rabi oscillation, and this case is resonant due to the degeneracy of the ``initial" Hamiltonian $\tilde{H}^{\prime}$. But the ``perturbed" Hamiltonian $H$ is nothing else than the SDWP Hamiltonian of equation (\ref{SchU}).
Now, equations (\ref{probell}) and (\ref{probar}) predict the ``flip-flop" between the stationary states $\psi_{R}(x)$ and $\psi_{L}(x)$, so that, if the system is initial prepared in the state
\begin{equation}
\psi(x,t=0)=\psi_{L}(x),
\end{equation}
then we will have a 100\% certainty to find it in state $\psi_{R}(x)$ at times $t=\frac{\pi}{\omega},\frac{3\pi}{\omega},\frac{5\pi}{\omega}\ldots$ and a 100\% certainty to find it in state $\psi_{L}(x)$ at times $t=\frac{2\pi}{\omega},\frac{4\pi}{\omega},\frac{6\pi}{\omega}\ldots$. And this last result is consistent with equation (\ref{II.F}). Thus, we are in the presence of two different (yet not contradictory) descriptions of one and the same phenomenon: if $H$ is considered an unperturbed Hamiltonian, with complete stationary solutions $\psi_{0}(x,t)$ and $\psi_{1}(x,t)$, then we have an ``oscillating" non-stationary solution $\psi_{L}(x,t)$. If, on the other hand, $H$ is considered to be the result of a perturbation acting on the degenerate Hamiltonian $H^{\prime}$, we then get flip-flops between the complete stationary solutions of $H^{\prime}$, that is: $\psi_{L}(x,t)$ and $\psi_{R}(x,t)$.
In this manner, we obtain the periodic transitions (the zero point Leggett-Caldeira oscillations) observed in so many experiments. Notice that this transitions occur in the absence of external fields, thus without emission or absorption.
| 3,660 | 17,658 |
en
|
train
|
0.37.3
|
\section{Thermal radiation}\label{sec:4}
Due to the fact that no quantum system can be completely isolated from its environment, in any realistic description the Schroedinger equation must be supplemented with terms that describe the interaction between the system and its surroundings. But there are very different ways to describe this interaction and its results, depending on the time and energy scales involved, and the complexity of the analysis. Here we shall discuss the absorption-induced transitions by which the system is thrown into high-lying energy levels, rendering the two-level model inapplicable. The main result from this discussion will be a limit on the temperature at which Caldeira-Leggett oscillations can be observed.
\subsection{Oscillations near resonance}
Now, oscillatory behavior is to be expected not only for the the resonant, exactly degenerate, Hamiltonian matrix $\mathbb{H}^{\prime}$. Indeed, it would not be realistic to expect Leggett-Caldeira oscillations only in perfectly isolated systems. Consider a harmonic perturbation of the SDWP matrix Hamiltonian $\mathbb{H}$ of equation (\ref{IV.A.a}), that is, a perturbative term of the general form
\begin{equation}
\mathbb{V}=\mathbb{A}\exp(i\omega^{\prime}t)+\mathbb{A}^{\dagger}\exp(-i\omega^{\prime}t),
\end{equation}
and let us focus on the particularly simple case for which
\begin{equation}\label{simple.1}
\mathbb{A}=A\left(\begin{array}{cc}
0&1\\
0&0\\
\end{array}\right),
\end{equation}
so that the perturbative term can be written down as
\begin{equation}
\mathbb{V}=A\left(\begin{array}{cc}
0&\exp(i\omega^{\prime}t)\\
\exp(-i\omega^{\prime}t)&0\\
\end{array}\right).
\end{equation}
We then again resort to the time-dependent perturbation formalism, and write
\begin{equation}
\psi(x,t)=c_{0}(t)\psi_{0}(x,t)+c_{1}(t)\psi_{1}(x,t)
\end{equation}
in order to obtain the equation
\begin{equation}\label{mtdpC}
i\hbar\frac{d}{dt}\left(\begin{array}{c}
c_{0}\\
c_{1}\\
\end{array}\right)(t)=\mathfrak{V}(t)\left(\begin{array}{c}
c_{0}\\
c_{1}\\
\end{array}\right)(t),
\end{equation}
where $\mathfrak{V}$, defined by
\begin{equation}
\mathfrak{V}(t)=\exp\Big(i\mathbb{H} t/\hbar\Big)\mathbb{V}(t)\exp\Big(-i\mathbb{H} t/\hbar\Big)
\end{equation}
represents the perturbation in the interaction picture, and in our particularly simple case reduces to
\begin{equation}
\mathfrak{V}(t)=A\left(\begin{array}{ccc}
0&\quad&\exp \left(i(\omega^{\prime}-\omega)t \right)\\
& \\
\exp \left(-i(\omega^{\prime}-\omega)t \right) &\quad&0\\
\end{array}\right),
\end{equation}
so that equation (\ref{mtdpC}) is equivalent to the $2 \times 2$ system of coupled ODEs
\begin{equation}
i\hbar\frac{dc_{0}}{dt}=A\exp\Big[ i\big(\omega^{\prime}-\omega\big)t\Big] c_{1}\textrm{,}\quad
i\hbar\frac{dc_{1}}{dt}=A\exp\Big[ -i\big(\omega^{\prime}-\omega\big)t\Big] c_{0}\textrm{.}
\end{equation}
It can be checked by hand that
\begin{equation}
c_{0}(t)=\exp(i\Omega^{\prime} t/2)\Bigg\{\cos (R_{0} t)-\frac{i\Omega^{\prime}}{2R_{0}}\sin (R_{0}t)\Bigg\}
\end{equation}
and
\begin{equation}
c_{1}(t)=\frac{-iR_{1}}{R_{0}}\exp(-i\Omega^{\prime} t/2)\sin (R_{0}t)
\end{equation}
provide a solution for the initial conditions $c_{0}(t=0)=1 , c_{1}(t=0)=0$. Here we have used the following shorthand
\begin{equation}\begin{array}{cccc}
R_{0}=\sqrt{(A/\hbar)^{2}+\Big(\frac{\omega^{\prime}-\omega}{2}\Big)^{2}},&\ \Omega^{\prime}=\omega^{\prime}-\omega &
\textrm{ and }&
R_{1}=A/\hbar,\
\end{array}
\end{equation}
which lead to what is known as Rabi's formula,\cite{Fitzpatrick} namely:
\begin{equation}\label{Rah.1}
P_{1}(t)=\bigg(\frac{R_{1}}{R_{0}}\bigg)^{2}\sin^{2}\big(R_{0}t\big),
\end{equation}
\begin{equation}\label{Rah.2}
P_{0}(t)=1-P_{1}(t).
\end{equation}
It is not difficult to find the expressions for $P_{L}(t)$ and $P_{R}(t)$ for this particular choice of $\mathbb{A}$. We omit these, as they are not particularly illuminating. Let us just point out that in all instances $P_{R}$ and $P_{L}$ are oscillating functions of time, although they are generally not periodic. If one wishes to describe periodic Rabi oscillations in the $R$ and $L$ states, one should take, instead of (\ref{simple.1}),
\begin{equation}
\mathbb{A}=\frac{1}{2}\left(\begin{array}{cc}
1&-1\\
1&-1\\
\end{array}\right)
\end{equation}
as the natural choice for $\mathbb{A}$. By doing this one obtains expressions completely analogous to (\ref{Rah.1}) and (\ref{Rah.2}) for $P_{L}$ and $P_{R}$:
\begin{equation}
P_{L}(t)=\bigg(\frac{R_{1}}{R_{0}^{\prime}}\bigg)^{2}\sin^{2}\big(R_{0}^{\prime} t\big)
\end{equation}
\begin{equation}
P_{R}(t)=1-P_{L}(t),
\end{equation}
where the new frequency of the oscillation is now given by
\begin{equation}
R_{0}^{\prime}=\sqrt{\big(A/\hbar\big)^{2}+\big(\omega^{\prime}/2\big)^{2}}
\end{equation}
The main conclusion of this subsection is thus, that the Leggett-Caldeira can survive the influence of an environment on the particle in a SDWP under certain circumstances.
\subsection{A limit on temperature}
An important result from perturbation theory tells us that for harmonic perturbations the time-dependent transition amplitude, $c_{n\rightarrow m}(t)$, between two given eigenstates of the complete Hamiltonian $H$ is given by:\cite{EspositoMarmoSudarshan}
\begin{equation}\label{III.6}
c_{n\rightarrow m}(t)=\langle \psi_{m}\vert A\vert\psi_{n}\rangle\frac{1-\exp i(\frac{E_{m}-E_{n}}{\hbar}-\omega^{\prime})t}{E_{m}-E_{n}-\hbar\omega^{\prime}}+\langle \psi_{m}\vert A^{*}\vert\psi_{n}\rangle\frac{1-\exp i(\frac{E_{m}-E_{n}}{\hbar}+\omega^{\prime})t}{E_{m}-E_{n}+\hbar\omega^{\prime}}.
\end{equation}
As a consequence we get that, if the system is to stay in the two lowest lying levels, then the perturbation must meet the condition:
\begin{equation}
\omega^{\prime}<\frac{E_{2}-E_{1}}{\hbar}.
\end{equation}
Otherwise, the perturbation would excite the system to higher levels with non-negligible probability. This gives a limit on the temperature at which the system behaves like a low-lying two-level system. Indeed, recalling Wien's law for blackbody radiation, we get that thermal radiation at a temperature $T$ will have a maximal contribution of frequency $\omega^{\prime}$ when condition
\begin{equation}
\omega^{\prime}=\frac{2\pi c}{b_{W}}T
\end{equation}
is met. (Here, $T$ stands for the temperature of the radiation, $b_{W}$ is Wien's constant, and $c$ the velocity of light) Thus, if $\mathbb{V}(x,t)$ is somehow to represent thermal radiation, and if the perturbed Hamiltonian $H^{\prime}$ is to be described as a low-lying two-level system, then we must have:
\begin{equation}\label{temp.1}
T<\frac{b_{W}}{2\pi c}\ \frac{E_{2}-E_{1}}{\hbar}.
\end{equation}
In so many words: for each system there is a limit temperature above which the two-level system description is inapplicable, and zero-point Leggett-Caldeira oscillations become overshadowed by other transitions. Moreover, from inequalities (\ref{inek01}) and (\ref{2NF}) we get :
\begin{equation}\label{temp.2}
T_{B}(a,m)<\frac{b_{W}}{2\pi c}\ \frac{E_{2}-E_{1}}{\hbar}<3T_{B}(a,m)
\end{equation}
with this global bound given by:
\begin{equation}\label{temp.3}
T_{B}(a,m)=\frac{5\pi\hbar b_{W}}{16mca^{2}}
\end{equation}
The meaning of expressions (\ref{temp.2}) and (\ref{temp.3}) is the following: consider a family of double rectangular barriers, with a fixed $m$, $a$ and $k$, but free barrier width. When exposed to thermal radiation, there is a temperature $T_{B}$ for the radiation above which the Leggett-Caldeira oscillations are overshadowed by other transitions in least some the systems, and at temperature $3T_{B}$ the Calderia-Legget oscillations are surpassed by other transitions in all of the systems.
| 2,512 | 17,658 |
en
|
train
|
0.37.4
|
\section{Decoherence and dissipation}\label{sec:5}
We begin this section with a simplified exposition of mixed and pure states and the density matrix formalism as found in Landau and Lifshitz,\cite{LandauZ} to move on next to an also simplified rendering of some of Leggett's original argumentation. After that, decoherence is defined, an its relation with dissipation is briefly discussed.
The interaction of a system ($\mathfrak{S}$) with its surroundings ($\mathfrak{E}$) can be taken into account by considering a bigger \emph{isolated} system ($\mathfrak{U}$) which encompasses both $\mathfrak{S}$ and $\mathfrak{E}$ (that is: $\mathfrak{U}=\mathfrak{S}\bigcup\mathfrak{E}$). The state of this new, all including, system, $\mathfrak{U}$ is described by a state function $\Psi(j,\xi)$ that depends both on the coordinates of $\mathfrak{S}$ (the $j$) and the the coordinates of its environment (the $\xi$). The total Hamiltonian $H_{T}$ acting on $\mathfrak{U}$ can always be written in the form:
\begin{equation}
H_{T}=H+H_{\mathfrak{E}}+\lambda H_{I}
\end{equation}
where $H$ depends only on the $j$ and their generalized momenta, $H_{\mathfrak{E}}$ depends only on the $\xi$ and its momenta, and $H_{i}$ depends on both types of coordinates. We shall take the approximation, that $H$ is the Hamiltonian of $\mathfrak{S}$ when isolated, and that $H_{I}$ alone models the interaction between $\mathfrak{S}$ and $\mathfrak{E}$.
In principle, there can happy instances in which $\Psi(j,\xi)$ could be written as the product of two states functions:
\begin{equation}\label{pure}
\Psi(j,\xi)=\psi(j)\phi(\xi)
\end{equation}
but this does not need to be the case. States that can be written in the form (\ref{pure}) are called \emph{pure states} in the literature. States that are not pure are said to be \emph{mixed.}
In order to illustrate this let us consider the case in which both the original system and its surroundings can be represented as two-level systems. If the isolated Hamiltonian $H$ has eigenfunctions $\psi_{+}$ and $\psi_{-}$:
\begin{equation}
H\psi_{\pm}=E_{\pm}\psi_{\pm}
\end{equation}
and if $\phi_{\alpha}$ and $\phi_{\beta}$ are the eigenfunctions of $H_{e}$, \emph{i. e.}
\begin{equation}
H_{e}\phi_{\alpha}=E_{\alpha}\phi_{\alpha}\ , \ H_{e}\phi_{\beta}=E_{\beta}\phi_{\beta},
\end{equation}
then some examples of pure states are:
$$\begin{array}{ c }
\frac{1}{\sqrt{2}}(\psi_{+}\phi_{\beta}+\psi_{-}\phi_{\beta})=\frac{1}{\sqrt{2}}(\psi_{+}+\psi_{-})\phi_{\beta}, \ \frac{1}{2}\psi_{-}\phi_{\beta}+\frac{\sqrt{3}}{2}\psi_{-}\phi_{\alpha}=\psi_{-}(\frac{1}{2}\phi_{\beta}+\frac{\sqrt{3}}{2}\phi_{\alpha})\\
\textrm{and}\\
\frac{1}{4}\psi_{-}\phi_{\beta}+\frac{\sqrt{3}}{4}\psi_{-}\phi_{\alpha}-\frac{\sqrt{3}}{4}\psi_{-}\phi_{\beta}-\frac{3}{4}\psi_{+}\phi_{\alpha}=(\frac{1}{2}\psi_{-}-\frac{\sqrt{3}}{2}\psi_{+})(\frac{1}{2}\psi_{\beta}+\frac{\sqrt{3}}{2}\phi_{\alpha}).\\
\end{array}$$
On the other hand, as instances of mixed states, we can provide the following:
\begin{equation}\label{mixed}
\frac{1}{\sqrt{2}}(\psi_{-}\phi_{\alpha}+\psi_{+}\phi_{\beta})\ , \ \frac{1}{\sqrt{2}}(\psi_{-}\phi_{\beta}+\psi_{+}\phi_{\alpha})\ , \ \textrm{ and } \frac{1}{\sqrt{3}}(\psi_{+}\phi_{\alpha}+\psi_{+}\phi_{\beta}+\psi_{-}\psi_{\alpha}).
\end{equation}
The density matrix formalism was developed to treat systems that (like $\mathfrak{U}$) can present mixed states. The density matrix $\rho$ allows us to calculate the expected value $\langle f \rangle$ of any observable $f(x,p_{x})$ that depends only on the coordinates and momenta of $\mathfrak{S}$:
\begin{equation}\label{Tr}
\langle f \rangle\ =\ \textrm{Tr}\Big( f\rho\Big).
\end{equation}
The elements of the density matrix $\rho$ of a state $\Psi(j,\xi)$ of are defined as:
\begin{equation}\label{rho}
\rho_{j,j^{\prime}}=S_{\xi}\Psi^{*}(j,\xi)\Psi(j^{\prime},\xi),
\end{equation}
where $S_{\xi}$ stands for the sum over the discrete $\xi$ (if any) plus an integral over the continuous $\xi$ (if any). In the case of our $2\times 2$-level system, expression (\ref{rho}) reduces to
\begin{equation}\label{rho}
\rho_{j,j^{\prime}}=\Psi_{j,\alpha}^{*}\Psi_{j^{\prime},\alpha}+\Psi_{j,\beta}^{*}\Psi_{j^{\prime},\beta}, \quad j,j^{\prime}=\pm\ .
\end{equation}
The diagonal elements of density matrix, of the form $\rho_{j,j}$, are called \emph{populations,} while the off-diagonal elements (\emph{i. e.} the elements with $j\neq j^{\prime}$) are known as \emph{coherences.}
Suppose now that the 50-50 linear combinations
\begin{equation}
\psi_{L}=\frac{1}{\sqrt{2}}\Big(\psi_{+}+\psi_{-}\Big), \ \psi_{R}=\frac{1}{\sqrt{2}}\Big(\psi_{+}-\psi_{-}\Big)
\end{equation}
are eigenfunctions of a macroscopic observable $M$, let us say:
\begin{equation}
M\psi_{L,R}=\mu_{L,R} \psi_{L,R},
\end{equation}
and take then the mixed state given by
\begin{equation}\label{momix}
\Psi=c_{L}\psi_{L}\phi_{\alpha}+c_{R} \psi_{R}\phi_{\beta}.
\end{equation}
The density matrix associated with (\ref{momix}) is written as
\begin{equation}\label{diag}
\rho=\left(\begin{array}{cc}
\vert c_{L}\vert^{2}& 0\\
0&\vert c_{R}\vert^{2}\\
\end{array}\right)
\end{equation}
in the $\{\psi_{L}, \psi_{R}\}$ basis, as can be seen from (\ref{rho}), so that according to (\ref{Tr}) the expected value of any observable $f$ pertaining to $\mathfrak{S}$ yields the value
\begin{equation}\label{expvalue}
\langle f\rangle=\vert c_{L}\vert^{2}f_{L}+\vert c_{R}\vert^{2}f_{R},
\end{equation}
where $f_{L}$ and $f_{R}$ are the expected values of $f$ in the pure states
\begin{equation}
\Psi_{L}=\psi_{L}\phi_{\alpha}\ \textrm{ and }\ \Psi_{R}=\psi_{R}\phi_{\beta}.
\end{equation}
The point of this discussion is that the same result (\ref{expvalue}) is obtained if we make measurements on an ensemble of $\mathfrak{U}$ systems all in state $\Psi$, or if the same measurements are made with an ensemble of $\mathfrak{U}$ made up of a statistical mixture of the pure states $\Psi_{L}$ and $\Psi_{R}$, in proportions $\vert c_{L}\vert^{2}$ and $\vert c_{R}\vert^{2}$. If it were to be held true that only ensembles of the type (\ref{momix}) could be prepared for $\mathfrak{U}$, then it could be argued that property $M$ has a sharp value for each element of the ensemble, and that a measurement done on a particular element only removes our ignorance on its value for that particular system. Clearly, this opens the door for hidden variable theories. To put it succinctly: in this interpretation each one of the Schroedinger's cats in an ensemble of such felines would be either dead or alive, and never in superpositions composed of both dead and alive states. Only the behaviour of the ensemble would be quantal, its individual elements being essentially classical.
It is patent, on the other hand, that the pure state
\begin{equation}
\Psi_{+}=\psi_{+}\phi_{\alpha}
\end{equation}
cannot be written as a mixed state of the form (\ref{momix}) and that its corresponding density matrix cannot be diagonal in the $(L,R)$ basis, unlike (\ref{diag}). The impossibility of the simultaneous diagonalization of the density matrices of all possible states of a system $\mathfrak{S}$ is then a strong evidence of the true quantal behaviour of such system, as opposed to the behaviour required by hidden variable theories. Thus, for a system $S$ to be classical in any sense of the word, the coherences, \emph{i. e.} the off-diagonal elements, must be absent from the density matrix for each one of its possible states. This conclusion is generally valid, even if we resorted to the most trivial case in order to illustrate it.\cite{Leggett}
Decoherence can be defined as the decay of the off-diagonal elements in the density matrix as a result of the interaction of the system with its environment. Therefore decoherence allows a system to behave as quantal when isolated and as classical when the coupling with its environment is ``sufficiently effective." This is nowadays considered a plausible mechanism for the emergence of classical reality from a quantal substratum.
In most practical applications, the environment $\mathfrak{E}$ has a very large number of degrees of freedom (say of the order of the Avogadro number) and not just one, as in the example we have used. Thus $\mathfrak{U}$ is usually a thermodynamic system, so that the full toolbox of quantum statistical mechanics needs to be marshalled in order to describe its behaviour. In this case the interaction between $\mathfrak{S}$ and $\mathfrak{E}$ (interaction known as quantum dissipation in this context) involves the relaxation of the thermodynamical variables of $\mathfrak{U}$ towards thermal equilibrium, and not only decoherence.
Various models have been proposed over the years for the environment (or \emph{bath}) but one of first and most successful is the \emph{spin-boson Hamiltonian} approach, in which $H_{\mathfrak{E}}$ is taken as a collection of harmonic oscillators with various frequencies and the interaction term $H_{I}$ is linear both in the $j$ and in the $\xi$ coordinates. One important result from this approach is that a two-level system $\mathfrak{S}$ will describe damped oscillations between the localized states $\vert R\rangle$ and $\vert L\rangle$. Depending on the frequency distribution of the environment, $\mathfrak{S}$ may be localized at $T=0^{o}$K (the overdamped case, known as ``subohmic"), it may present critical damping (the ``ohmic case") or it may undergo underdamped coherent oscillations (the ``superohmic case.") The last one of these three instances is the most interesting for the present discussion, as it allows the observation of MQC before the complete relaxation of the system. The possibility of experimental MQC in the superohmic case depends in the interplay between a \emph{decoherence time} defined only by the bath parameters, and the period of the Leggett-Caldeira oscillation for system $\mathfrak{S}$.
| 2,856 | 17,658 |
en
|
train
|
0.37.5
|
\section{The scales of MQC}\label{sec:6}
\begin{table}[h!]
\centering
\begin{tabular}{|c|c|c|c|c|}
\hline
$b$ & $E_0$ & $E_1$ & $\Delta E$ & $\tau$ \\
( nm )& $( \times 10^{-26} $ J )& ( $\times 10^{-26}$ J ) & $ ( \times 10^{-28}$J )& ( $\mu$s )\\
\hline \hline
100.00000& 5.3753895& 5.4382093& 6.3 & 1.0\\ \hline
116.65290& 5.3899569& 5.4246062& 3.5 & 2.9 \\ \hline
136.07900& 5.3987829& 5.4160961& 1.7 & 3.8\\ \hline
158.74011& 5.4036276& 5.4113353& 0.77 & 8.6\\ \hline
185.17494& 5.4059909& 5.4089897& 0.30 & 22.0\\ \hline
216.01195& 5.4069931& 5.4079902& 0.10 & 66.0\\ \hline
251.98421& 5.4073539& 5.4076298& 2.7$\times 10^{-2}$ & 240\\ \hline
\hline
\end{tabular}
\caption{Period $\tau$ increases exponentially as the barrier width is augmented. $a=1.0\ \mu$m, $k=2\times10^{-20}$J. This table, as well as all figures, was generated with Matlab \textsuperscript{\textregistered} R2012a.}
\label{tab:sample}
\end{table}
Let us start by fixing the width of the lateral wells at:
\begin{equation}\label{VII.1}
a=1\mu\textrm{m,}
\end{equation}
a value typical of contemporary lithographic circuitry, and take $m$ to be the rest mass of an electron:
\begin{equation}\label{VII.2}
m=m_{e}=9.1\times 10^{-31}\textrm{kg}.
\end{equation}
With this, $B$ takes the value:
\begin{equation}\label{5.A}
B=0.6\times 10^{-25}\textrm{J}=0.36\ \mu\textrm{eV,}
\end{equation}
and $T_{B}$ is fixed at:
\begin{equation}
T_{B}\approx 1.1\ \textrm{mK.}
\end{equation}
From equation (\ref{II.7}), that gives the fundamental frequency of the Caldeira-Leggett oscillations, we get the corresponding period
\begin{equation}
\tau=\frac{2\pi\hbar}{E_{1}-E_{0}}.
\end{equation}
A global lower bound for this period is found from expressions (\ref{inequalityA}) and (\ref{funnyeq}):
\begin{equation}\label{5.B}
\tau >\frac{2\pi\hbar}{B}=\frac{4ma^{2}}{\pi\hbar}.
\end{equation}
For values (\ref{VII.1}) and (\ref{VII.2}) this gives
\begin{equation}
\tau>11\textrm{ns}.
\end{equation}
From table I (obtained through computer assisted numerical analysis) we get that as we sweep the barrier width from 0.2 to 0.5 $\mu$m the period of the Leggett -Caldeira oscillations for our square double well increases from 1.0 to 240 $\mu$s. Based on general considerations it has been estimated\cite{Leggett} that, for all practical purposes, MQC is lost if the period of the Legget-Caldiera oscillation is of the order $\tau\gtrsim100\mu$s. Thus, the last row of the table corresponds to a localized system. All the other tabulated values could in principle correspond to observable MQC.
\subsection{Some of the many things we have left out}
MQC experiments are carried out in superconducting quantum interference devices (SQUIDs) with low capacitance tunneling Josephson junctions,\cite{Leggett, Friedmanetal} and the relevant coordinate (\emph{i. e.} the analogous of coordinate $x$) is not of a geometric character (like a position) but is in most cases the phase difference between the states functions of the electrons in a Cooper pair (so that $m$ is not really the mass of the electron.) Thus our toy model is in reality a simplification of a mechanical analogy used to discuss experimental MQC.
\section{Conclusions}\label{sec:7}
Contemporary quantum mechanics, both experimental and theoretical, provides examples of basic concepts and techniques such as: tunneling, stationary states, two-level systems, perturbation theory, the density matrix and the WKB approximation. Classroom presentations of current areas of research, such as MQC, help to improve the understanding of quantum physics at university level, as they connect the simplified textbook models with the actual state of the field, and thus with the future professional activity of the student. Moreover, MQC illustrates in a beautiful way the interplay between theory and experiment, and between concepts and techniques arising in different areas of quantum physics.
We believe to have achieved in the present paper a level of exposition that makes it both clear and interesting for senior university students and recent graduates. In order to do so, we had to glide over the more technical aspects of experimental MQC and the intricate relation between MQC and the epistemology and the philosophy of physics. We hope that the present paper will encourage the interested reader to delve further into this facets of contemporary research.
\section{Appendix}
Consider condition (\ref{coneven}) for the ground level ($n=0$), that is:
\begin{equation}\label{app01}
E_{0}\cot^{2} a\frac{\sqrt{2mE_{0}}}{\hbar}=(k-E_{0})\tanh^{2}b\frac{\sqrt{2m(k-E_{0})}}{\hbar}
\end{equation}
We will now establish a lower bound for $E_{0}$ starting from (\ref{app01}), but we have to take some precautions in doing so because $E_{0}$ depends implictly on $b.$ In order to proceed, note that
\begin{equation}\label{app01A}
\forall b\in(0,\infty)\ , \ \quad\frac{\sqrt{2m(k-E_{0})}}{\hbar}<\frac{\sqrt{2m(k-B/4)}}{\hbar}
\end{equation}
so that
\begin{equation}\label{app02}
\forall b\in(0,\infty) \ , \ \quad \tanh^{2}b\frac{\sqrt{2m(k-E_{0})}}{\hbar}>\tanh^{2} b\frac{\sqrt{2m(k-B/4)}}{\hbar}
\end{equation}
The dependence of the rhs of inequality (\ref{app02}) is explicit, so that the usual procedures of calculus can be applied. In particular as we now from elementary theorems that the limit
\begin{equation}
\lim_{b\rightarrow \infty}\tanh^{2} b\frac{\sqrt{2m(k-B/4)}}{\hbar}=1
\end{equation}
holds true, we can affirm that: for given $\delta>0$ there exists a $b_{0}( \delta)$ such that any $b>b_{0}(\delta)$
\begin{equation}\label{app03}
\tanh^{2} b\frac{\sqrt{2m(k-B/4)}}{\hbar}>1-\frac{\delta}{2k}
\end{equation}
From (\ref{app01}) (\ref{app02}) and (\ref{app03}) we deduce that for any $b$ above a certain value $b_{0}(\delta)$, the ground energy of $U_{b}$ satisfies:
\begin{equation}\label{app04}
E_{0}\cot^{2}a\frac{\sqrt{2mE_{0}}}{\hbar}>(k-E_{0})\Big(1-\frac{\delta}{2k}\Big)
\end{equation}
Turning our attention to the condition for $E_{1}$, \emph{i. e.}
\begin{equation}\label{app01B}
E_{1}\cot^{2} a\frac{\sqrt{2mE_{1}}}{\hbar}=(k-E_{1})\coth^{2}b\frac{\sqrt{2m(k-E_{1})}}{\hbar}
\end{equation}
we now find an upper bound for $E_{1}$, by noting that, because of (\ref{app01A}) and the known properties of the hyperbolic functions, the inequality
\begin{equation}
\coth^{2}b\frac{\sqrt{2m(k-E_{1})}}{\hbar}<\coth^{2}b\frac{\sqrt{2m(k-B/4)}}{\hbar}
\end{equation}
is verified for all strictly positive $b$. Furthermore,
\begin{equation}
\lim_{b\rightarrow\infty}\coth^{2}b\frac{\sqrt{2m(k-B/4)}}{\hbar}=1
\end{equation}
so that for every $\delta>0$ there exist a $b_{1}(\delta)$ such that, if $b>b_{1}(\delta)$, then inequality
\begin{equation}\label{app07A}
\coth^{2}b\frac{\sqrt{2m(k-B/4)}}{\hbar}<1+\frac{\delta}{2k}
\end{equation}
is satisfied for all strictly positive $b.$ And from (\ref{app01B}) and (\ref{app07A}) we get that, for all $b$ above a certain thershold value $b_{1}(\delta)$, the inequality
\begin{equation}\label{app08}
E_{1}\cot^{2}a\frac{\sqrt{2mE_{1}}}{\hbar}<(k-E_{1})\Big(1+\frac{\delta}{2k}\Big)
\end{equation}
is satisfied.
Taking both (\ref{app04}) and (\ref{app08}) into consideration, we have that for every $\delta >0$ there exists a number $b(\delta)=\max \{b_{0}(\delta) b_{1}(\delta)\}$ such that for any $b>b(\delta)$ the inequality
\begin{equation}\label{app09}
E_{1}\cot^{2}a\frac{\sqrt{2mE_{1}}}{\hbar}-E_{0}\cot^{2}a\frac{\sqrt{2mE_{0}}}{\hbar}<(E_{0}-E_{1})+\delta \Big(1-\frac{E_{0}+E_{1}}{2k}\Big)
\end{equation}
is satisfied. Now, it is not difficult to see that
\begin{equation}
w(E)=E\cot^{2}a\frac{\sqrt{2mE}}{\hbar}
\end{equation}
is a monotonically increasing function of $E$ in the range $B/4<E<B$, so that
\begin{equation}\label{app010}
0<E_{1}\cot^{2}a\frac{\sqrt{2mE_{1}}}{\hbar}-E_{0}\cot^{2}a\frac{\sqrt{2mE_{0}}}{\hbar}
\end{equation}
and in the other hand, we deduce
\begin{equation}\label{app011}
(E_{0}-E_{1})+\delta \Big(1-\frac{E_{0}+E_{1}}{2k}\Big)<\Big(1-\frac{E_{0}+E_{1}}{2k}\Big)\delta <\delta
\end{equation}
from and . From ( \ref{app09}), (\ref{app010}) and (\ref{app011}) we get:
\begin{equation}
0<E_{1}\cot^{2}a\frac{\sqrt{2mE_{1}}}{\hbar}-E_{0}\cot^{2}a\frac{\sqrt{2mE_{0}}}{\hbar}<\delta
\end{equation}
\begin{equation}
E_{1}\cot^{2} a\frac{\sqrt{2mE_{1}}}{\hbar}>k-E_{1}
\end{equation}
Finally, we notice that, as
\begin{equation}
v(E)=\cot^{2}a\frac{\sqrt{2mE}}{\hbar}
\end{equation}
is a monotonically increasing function of $E$ in the range $B/4<E<B$, then
\begin{equation}\label{app013}
(E_{1}-E_{0})\cot^{2}a\frac{\sqrt{2mE_{0}}}{\hbar}<\delta
\end{equation}
Now we just need to find a lower bound on $\cot^{2}a\frac{\sqrt{2mE_{0}}}{\hbar}$. This is obtained by turning back to condition (\ref{app01}) from which we get
\begin{equation}\label{app014}
\cot^{2}a\frac{\sqrt{2mE_{0}}}{\hbar}<\frac{k-B/4}{B}
\end{equation}
Finally, from (\ref{app013}) and (\ref{app014}) we arrive at
\begin{equation}\label{app015}
E_{1}-E_{0}<\delta\frac{B}{k-B/4}
\end{equation}
Let us stress that $k$ and $B$ are independent of $b$. In this manner, we have arrived at the following lemma:
For each strictly positive real number $ \delta$ there exists a
\begin{equation}
b^{\prime}(\delta)=b(\delta\frac{k-B/4}{B})
\end{equation}
such that for any $b>b^{\prime}(\delta)$ the gap between the ground and first excited levels of of $U_{b}$ is less than $\delta$, that is, such that:
\begin{equation}
E_{1}-E_{0}<\delta \ .
\end{equation}
And this is what we set out to prove in this appendix.
\end{document}
| 3,610 | 17,658 |
en
|
train
|
0.38.0
|
\begin{document}
\title{On the Schr\"{o}dinger equation with singular potentials}
\author{{{Jaime Angulo Pava}}\\{\small IME-USP, Rua do Matao 1010, Cidade Universitaria}\\{\small {CEP 05508-090, Sao Paulo, SP, Brazil.}}\\{\small \texttt{E-mail:[email protected]}}
\\{{Lucas C. F. Ferreira}}\\{\small Universidade Estadual de Campinas, IMECC - Departamento de
Matem\'{a}tica,} \\{\small {Rua S\'{e}rgio Buarque de Holanda, 651, CEP 13083-859, Campinas-SP,
Brazil.}}\\{\small \texttt{E-mail:[email protected]}}}
\date{}
\maketitle
\begin{abstract}
We study the Cauchy problem for the non-linear Schr\"odinger equation with singular
potentials. For point-mass potential and nonperiodic case, we prove existence
and asymptotic stability of global solutions in weak-$L^{p}$ spaces. Specific interest is give to the point-like $\delta$ and $\delta'$ impurity and for two $\delta$-interactions in one dimension. We also consider the
periodic case which is analyzed in a functional space based on Fourier transform and
local-in-time well-posedness is proved.
\end{abstract}
{\small {\quad\textbf{AMS subject classification:} 35Q55, 35A05, 35A07, 35C15, 35B40, 35B10 }}
{\small
\quad\textbf{Keywords} NLS-Dirac equation, Singular
potential, Existence, Asymptotic behavior}
| 460 | 34,109 |
en
|
train
|
0.38.1
|
\section{Introduction}
We are interested in this paper in the the Cauchy problem for the following Schr\"odinger model
\begin{equation}
\left\{
\begin{aligned}
i\partial_{t}u+\Delta u+\mu(x)u & = F(u),\quad x\in \mathbb R^n, \;\;t\in \mathbb R\ \label{SCH0}\\
u(x,0) & =u_{0}(x),
\end{aligned}\right.
\end{equation}
in weak-$L^p$ spaces (Marcinkiewicz spaces) and in a space based on Fourier transform. In the weak-$L^p$ spaces we consider the case $n=1$ and $\mu(x)=\sigma \delta$, $\mu(x)=\sigma(\delta(\cdot-a)+\delta(\cdot+a))$ (two Dirac's $\delta$ potentials place at the points $\pm a \in \mathbb R$) or $\mu(x)=\sigma \delta'$ where $\delta$ represents the delta function in the origin and $\sigma\in \mathbb R$, $F(u)= \lambda\left\vert u\right\vert ^{\rho
-1}u$, where $\lambda=\pm1$ and $\rho>1$. In the space based on Fourier transform we consider $n$ arbitrary and $\mu(x)$ being a bounded continuous function with a Fourier transform being a finite Radon measure and $F(u)= \lambda u^{\rho}$, where $\lambda=\pm1$ and $\rho\in \mathbb N$. The case $F(u)= \lambda\left\vert u\right\vert ^{\rho
-1}u$ is also commented.
The non-linear Schr\"odinger model (\ref{SCH0}) in the case $\mu(x)=\sigma\delta(x)$ (called the non-linear Schr\"odinger equation with a $\delta$-type impurity, the NLS-$\delta$ equation henceforth) arise in different areas of quantum field theory and are essential for understanding a number of phenomena in condensed matter physics. At the experimental side, the recent interest in point-like impurities (defects) is triggered by the great progress in building nanoscale devises. More exactly, the NLS-$\delta$ model with a impurity at the origin in the repulsive ($\sigma <0$) case and in the attractive ($\sigma >0$) is described by the following boundary problem (see Caudrelier\&Mintchev\&Ragoucy \cite{CMR})
\begin{equation}
\left \{
\begin{aligned}
i\partial_{t}u(x,t)+ u_{xx}(x,t) & = \lambda \left\vert u(x,t)\right\vert ^{\rho
-1}u(x,t),\quad x\neq0 \\
\lim_{x\to 0^+}[u(x,t)-u(-x, t)]&=0, \\
\lim_{x\to 0^+}[\partial_x u(x,t)-\partial_x u(-x, t)]&=\sigma u(0,t) \label{SCH02}\\
\lim_{x\to \pm \infty} u(x,t)=0,
\end{aligned}\right.
\end{equation}
hence $u(x,t)$ must be solution of the non-linear Schr\"odinger equation on $\mathbb R^{-}$ and $\mathbb R^{+}$, continuous at $x=0$ and satisfy a ``jump condition'' at the origin and it also vanishes at infinity.
The equations in (\ref{SCH02}) are a particular case of a more general model considering that the impurity is localized at $x=0$; in fact the equation of motion
$$
i\partial_{t}u(x,t)+ u_{xx}(x,t) = \lambda \left\vert u(x,t)\right\vert ^{\rho
-1}u(x,t),\quad x\neq0,
$$
with the impurity boundary conditions
\begin{equation}\label{bc}
\left(\begin{array}{c}u(0+,t) \\ \partial_xu(0+,t)\end{array}\right)=\alpha
\left(\begin{array}{cc} a & b\ \\c & d\end{array}\right)\left(\begin{array}{c} u(0-,t) \\ \partial_xu(0-,t) \end{array}\right)
\end{equation}
with
\begin{equation}\label{para}
\{a,b,c,d\in \mathbb R, \alpha\in \mathbb C: ad-bc=1, |\alpha |=1\}.
\end{equation}
The equation (\ref{bc}) captures the interaction of the ``field'' $u$ with the impurity \cite{CMR2}. The parameters
in (\ref{para}) label the self-adjoint extensions of the (closable) symmetric operator $H_0=-\frac{d^2}{dx^2}$ defined on the space $C_0^{\infty}(\mathbb R-\{0\})$ of smooth functions with compact support separated from the origin $x=0$. In fact, by von Neumann-Krein's theory of self-adjoint extensions for symmetric operators on Hilbert spaces, it is not difficult to show that there is a 4-parameter family of self-adjoint operators which describes all one point interactions in one-dimension of the second derivative operator $H_0$. Such a family
can be equivalently described through the family of boundary conditions at the origin
\begin{equation}\label{bc1}
\left(\begin{array}{c}\psi(0+) \\ \psi'(0+)\end{array}\right)=\alpha
\left(\begin{array}{cc} a & b\ \\c & d\end{array}\right)\left(\begin{array}{c} \psi(0-) \\ \psi'(0-)\end{array}\right)
\end{equation}
with $a,b,c,d$ and $\alpha$ satisfying the conditions in (\ref{para}) (see Theorem 3.2.3 in \cite{ak}).
Here we are interested in two specific choices of the parameters in (\ref{para}), which are relevant in physics applications (see \cite{CMR2}-\cite{CMR}). The first choice $\alpha=a=d=1$, $b=0$, $c=\sigma\neq 0$ corresponds to the case of a pure Dirac $\delta$ interaction of strength $\sigma$ (see Theorem \ref{self} below). The second one $\alpha=a=d=1$, $c=0$, $b=\beta\neq 0$ corresponds to the case of the so-called $\delta'$ interaction of strength $\beta$ (see Theorem \ref{selfd} below).
In section 2 below for convenience of the reader we present a precise formulation for the point interaction determined by the formal linear differential operator
\begin{equation}\label{deltaop0}
-\Delta_{\sigma}=-\frac{d^2}{dx^2}+\sigma \delta,
\end{equation}
which will be match with the singular boundary condition in (\ref{SCH02}) at every time $t$.
Existence and uniqueness of local and global-in-time solutions of problem (\ref{SCH0}) with $\mu(x)=0$ and $F(u)= \lambda\left\vert u\right\vert ^{\rho-1}u$ have been much studied in the framework of the Sobolev spaces $H^s(\mathbb R^n)$, $s\geqq 0$, i.e, the solutions and their derivatives have finite energy (see Cazenave's book \cite{C1} and the reference therein). In the case of $\delta$-interaction, namely, $\mu(x)=\sigma \delta$ the existence of global solution in $H^1(\mathbb R)$ and $L^2(\mathbb R)$ has been addressed in Adami\&Noja \cite{Adami} (we can also to apply Theorem 3.7.1 in \cite{C1} for obtaining a local-in-time well-posedness theory in $H^1(\mathbb R)$).
The first study of infinite $L^2$-norm solutions for $\mu(x)=0$ and $F(u)=\lambda\left\vert u\right\vert ^{\rho-1}u$ was addressed by Cazenave\&Weissler in \cite{Cazenave1} where they consider the space
$$
X_\rho=\{u:\mathbb{R}\rightarrow L^{\rho+1}(\mathbb R^n) \text{ \ Bochner meas.}: \sup_{-\infty<t<\infty}|t|^{\vartheta }\|u(t)\|_{L^{\rho+1}}<\infty\},
$$
where $\vartheta =\frac{1}{\rho-1}-\frac{n}{2(\rho+1)}$ and $\|\cdot\|_{L^{\rho+1}}$ denotes the usual ${L^{\rho+1}}$ norm. Under a suitable smallness condition on the initial data, they prove the existence of global solution in $X_\rho$, for $\rho_{0}(n)<\rho<\gamma(n)$ where $\rho_{0}(n)=\frac{n+2+\sqrt{n^{2}+12n+4}}{2n}>1$ is the positive root of the equation $n\rho^{2}-(n+2)\rho-2=0$ and $\gamma(n)=\infty$ if $n=1,2$ and $\gamma(n)=$ $\frac{n+2}{n-2}$ in otherwise.
Later on, in Cazenave\&Vega\&Vilela \cite{Cazenave2} the Cauchy problem was studied in the framework of weak-$L^p$ spaces. Using a Strichartz-type inequality, the authors obtained existence of solutions in the class $L^{(p,\infty)}(\mathbb R^{n+1})\equiv L_t^{(p,\infty)}(L_x^{(p,\infty)})$, where $(x,t)\in \mathbb R^{n}\times \mathbb R$ and $p=\frac{(\rho-1)(n+2)}{2\rho}$, for $\rho$ in the range
$$
\rho_0<\frac{4(n+1)}{n(n+2)}<\rho-1<\frac{4(n+1)}{n^2}<\frac{4}{n-2}.
$$
More recently, in Braz e Silva\&Ferreira\&Villamizar-Roa \cite{BFV} the Cauchy problem was studied in the Marcinkiewicz space $L^{(\rho+1,\infty)}$. Using bounds for the Schr\"odinger linear group in the context of Lorentz spaces, the authors showed existence and uniqueness of local-in-time solutions in the class
$$
\{u:\mathbb{R}\rightarrow L^{(\rho+1,\infty)} \text{ \ Bochner
meas.}: \sup_{-T<t<T}|t|^{\zeta}\|u(t)\|_{L^{(\rho+1,\infty)}}<\infty\},
$$
where $1<\rho<\rho_{0}(n)$ and $\frac{n(\rho-1)}{2(\rho+1)}=\zeta_{0}<\zeta<\frac{1}{\rho}$.
Since $\rho_{0}(n)<\frac{4}{n}$, the range for $\rho$ is different from the ones in Cazenave\&Weissler \cite{Cazenave1} and Cazenave {\it et al.} \cite{Cazenave2}. The existence of global solutions is showed in norms of type $\sup_{|t|>0}|t|^{\vartheta }\|u(t)\|_{L^{(\rho+1,\infty)}}$, where $\vartheta =\frac{1}{\rho-1}-\frac{n}{2(\rho+1)}$ and $\rho_0(n)<\rho<\gamma(n)$.
Our approach is based in some ideas in \cite{BFV}, so via real interpolation techniques we establish bounds for the Schr\"odinger linear group $G_\sigma(t)=e^{i(\partial_x^2+\sigma \delta)t}$ in the context of Lorentz spaces in the one-dimensional case. The cases $n=2,3$ remain open. The fundamental solution of the corresponding linear time-dependent Schr\"odinger equation, namely
$$
iu_t=-(\Delta+\sigma \delta)u,
$$
is now well know for $ n=1,2,3$; see Albeverio {\it et al.} \cite{ABD}- for instance. However, surprisingly, a \textquotedblleft good formula\textquotedblright of the unitary group $G_\sigma(t)\phi=e^{i(\Delta+\sigma \delta)t}\phi$ depending of the free linear propagator $e^{i\Delta t}\phi$ was found explicitly only for the one-dimensional case (see Holmer {\it et al.} \cite{Holmer5}-\cite{Holmer3}). In fact, by using scattering techniques, it was established in \cite{Holmer5} the convenient formula (for the case $\sigma\geqq 0$)
\begin{equation}\label{pospro1}
G_\sigma(t) \phi(x)= e^{it \Delta} (\phi\ast \tau_\sigma) (x) \chi^0_{+} + \Big[e^{it \Delta} \phi(x) + e^{it \Delta} (\phi\ast \rho_\sigma) (-x) \Big ]\chi^0_{-}
\end{equation}
where
$$
\rho_\sigma(x)=-\frac{\sigma}{2} e^{\frac{\sigma}{2} x}\chi^0_{-},\;\; \tau_\sigma(x)=\delta (x)+ \rho_\sigma(x),
$$
with $\chi^0_{+}$ the characteristic function of $[0,+\infty)$ and $\chi^0_{-}$ the characteristic function of $(-\infty, 0]$. For the case $\sigma<0$ see \cite{Holmer3} and Theorem \ref{expli} below. Here we show in the Appendix how to obtain the formula (\ref{pospro1}) based in the fundamental solution found in Albeverio {\it et al.} \cite{ABD}, which does not use scattering ideas. Nice formulas as that in (\ref{pospro1}) are not known for the cases $n=2,3$. Now, in the case $\sigma\geqq 0$, one can show from (\ref{pospro1}) the dispersive estimate in Lorentz spaces (see Lemma \ref{grupint1} below)
\begin{equation}\label{grupint02}
\left\Vert G_\sigma(t)f\right\Vert _{(p^{\prime},d)}\leq C|t|^{-\frac
{1}{2}(\frac{2}{p}-1)}\left\Vert f\right\Vert _{(p,d)},
\end{equation}
for $1\leqq d\leqq \infty$, $p'\in (2,\infty)$, $p\in (1,2)$ and $p'$ satisfying $\frac{1}{p}+\frac{1}{p'}=1$, where $C>0$ is independent of $f$ and $t\neq 0$. Then, under a suitable smallness condition on the initial data $u_0$, the existence of global solutions for (\ref{SCH0}) is proved in the space (see Theorem \ref{GlobalTheo})
$$
\mathcal{L}_{\vartheta }^{\infty}=\{u:\mathbb{R}\rightarrow L^{(\rho+1,\infty)} \text{ \ Bochner
meas.}: \sup_{-\infty<t<\infty}|t|^{\vartheta }\Vert u\Vert_{(\rho+1,\infty)}<\infty\},
$$
for $\sigma\geq0$, where $\vartheta =\frac{1}{\rho-1}-\frac{1}{2(\rho+1)}$ and $\rho_{0}=\frac{3+\sqrt{17}}{2}>1$ is the positive root of the equation $\rho^{2}-3\rho-2=0$. We also analyze the asymptotic stability of the global solutions (see Theorem \ref{TeoAssin}). For $\sigma <0$ our approach in general is not applicable because in this case the operator $-\Delta_\sigma$ has a non-trivial negative point spectrum. But, in this case it is possible to show the existence of a invariant manifold of periodic orbits in Lorentz spaces (see Section 6).
With regard to the more two singular cases: two Dirac's $\delta$ potentials placed at the points $\pm a \in \mathbb R$, $\mu(x)=\alpha(\delta(\cdot-a)+\delta(\cdot+a))$, and $\mu(x)=\beta \delta'$, i.e., the derivative of a $\delta$, a similar analysis to that above for the case of a $\delta$-potential can be established. In these cases, it is not known an explicit expression for the associated time propagator as that in (\ref{pospro1}) for the case of $\mu(x)=\sigma\delta$. However, by using a formula for the integral kernel of the time propagator associated (see \cite{KS} and \cite{AGHH}), we obtain an estimate similar to (\ref{grupint02}).
| 4,031 | 34,109 |
en
|
train
|
0.38.2
|
Our approach is based in some ideas in \cite{BFV}, so via real interpolation techniques we establish bounds for the Schr\"odinger linear group $G_\sigma(t)=e^{i(\partial_x^2+\sigma \delta)t}$ in the context of Lorentz spaces in the one-dimensional case. The cases $n=2,3$ remain open. The fundamental solution of the corresponding linear time-dependent Schr\"odinger equation, namely
$$
iu_t=-(\Delta+\sigma \delta)u,
$$
is now well know for $ n=1,2,3$; see Albeverio {\it et al.} \cite{ABD}- for instance. However, surprisingly, a \textquotedblleft good formula\textquotedblright of the unitary group $G_\sigma(t)\phi=e^{i(\Delta+\sigma \delta)t}\phi$ depending of the free linear propagator $e^{i\Delta t}\phi$ was found explicitly only for the one-dimensional case (see Holmer {\it et al.} \cite{Holmer5}-\cite{Holmer3}). In fact, by using scattering techniques, it was established in \cite{Holmer5} the convenient formula (for the case $\sigma\geqq 0$)
\begin{equation}\label{pospro1}
G_\sigma(t) \phi(x)= e^{it \Delta} (\phi\ast \tau_\sigma) (x) \chi^0_{+} + \Big[e^{it \Delta} \phi(x) + e^{it \Delta} (\phi\ast \rho_\sigma) (-x) \Big ]\chi^0_{-}
\end{equation}
where
$$
\rho_\sigma(x)=-\frac{\sigma}{2} e^{\frac{\sigma}{2} x}\chi^0_{-},\;\; \tau_\sigma(x)=\delta (x)+ \rho_\sigma(x),
$$
with $\chi^0_{+}$ the characteristic function of $[0,+\infty)$ and $\chi^0_{-}$ the characteristic function of $(-\infty, 0]$. For the case $\sigma<0$ see \cite{Holmer3} and Theorem \ref{expli} below. Here we show in the Appendix how to obtain the formula (\ref{pospro1}) based in the fundamental solution found in Albeverio {\it et al.} \cite{ABD}, which does not use scattering ideas. Nice formulas as that in (\ref{pospro1}) are not known for the cases $n=2,3$. Now, in the case $\sigma\geqq 0$, one can show from (\ref{pospro1}) the dispersive estimate in Lorentz spaces (see Lemma \ref{grupint1} below)
\begin{equation}\label{grupint02}
\left\Vert G_\sigma(t)f\right\Vert _{(p^{\prime},d)}\leq C|t|^{-\frac
{1}{2}(\frac{2}{p}-1)}\left\Vert f\right\Vert _{(p,d)},
\end{equation}
for $1\leqq d\leqq \infty$, $p'\in (2,\infty)$, $p\in (1,2)$ and $p'$ satisfying $\frac{1}{p}+\frac{1}{p'}=1$, where $C>0$ is independent of $f$ and $t\neq 0$. Then, under a suitable smallness condition on the initial data $u_0$, the existence of global solutions for (\ref{SCH0}) is proved in the space (see Theorem \ref{GlobalTheo})
$$
\mathcal{L}_{\vartheta }^{\infty}=\{u:\mathbb{R}\rightarrow L^{(\rho+1,\infty)} \text{ \ Bochner
meas.}: \sup_{-\infty<t<\infty}|t|^{\vartheta }\Vert u\Vert_{(\rho+1,\infty)}<\infty\},
$$
for $\sigma\geq0$, where $\vartheta =\frac{1}{\rho-1}-\frac{1}{2(\rho+1)}$ and $\rho_{0}=\frac{3+\sqrt{17}}{2}>1$ is the positive root of the equation $\rho^{2}-3\rho-2=0$. We also analyze the asymptotic stability of the global solutions (see Theorem \ref{TeoAssin}). For $\sigma <0$ our approach in general is not applicable because in this case the operator $-\Delta_\sigma$ has a non-trivial negative point spectrum. But, in this case it is possible to show the existence of a invariant manifold of periodic orbits in Lorentz spaces (see Section 6).
With regard to the more two singular cases: two Dirac's $\delta$ potentials placed at the points $\pm a \in \mathbb R$, $\mu(x)=\alpha(\delta(\cdot-a)+\delta(\cdot+a))$, and $\mu(x)=\beta \delta'$, i.e., the derivative of a $\delta$, a similar analysis to that above for the case of a $\delta$-potential can be established. In these cases, it is not known an explicit expression for the associated time propagator as that in (\ref{pospro1}) for the case of $\mu(x)=\sigma\delta$. However, by using a formula for the integral kernel of the time propagator associated (see \cite{KS} and \cite{AGHH}), we obtain an estimate similar to (\ref{grupint02}).
For the case $n\geqq 4$ we do not have Schr\"odinger operators with point interactions. In fact, the Schr\"odinger operators with point interactions, namely, perturbations of the Laplace operators by ``measures'' supported on a discrete set (supported at zero for simplicity, namely, by the Dirac delta measure $\delta$ centered at zero) are usually defined by means of von Neumann\&Krein theory of self-adjoint extensions of symmetric operators, and so as one of a whole family of self-adjoint (in $L^2(\mathbb R^n)$) extensions of an operator $A$, $D(A)=C^\infty_0(\mathbb R^n-\{0\})$, $Au=-\Delta u$, $u\in D(A)$. In the case $n\geqq 4$ it is well known that the theory trivializes where there is only one self-adjoint extension of $A$ (see Albeverio {\it et al.} \cite{AGHH}).
Next, let $\mathcal M$ be the set of finite Radon measure endowed with the norm of total variation, that is, $\|\omega\|_{\mathcal M}=|\omega|(\mathbb R^n)$ for $\omega\in \mathcal M$, $n\geq1$. Then, by considering $\mu(x)$ in (\ref{SCH0}) being a bounded continuous function with a Fourier transform such that $\widehat{\mu}\in \mathcal M$, we show a local-in-time well-posedness result in the Banach space
\begin{equation}
\mathcal{I}=[\mathcal{M}(\mathbb{R}^{n})]^{\vee}=\{f\in\mathcal{S}^{\prime
}(\mathbb{R}^{n}):\widehat{f}\in\mathcal{M}(\mathbb{R}^{n})\}
\end{equation}
whose norm is given by $\|f\|_\mathcal I=\|\widehat{f}\|_{\mathcal M}$. We also obtain a similar result in the periodic case (see Section 7).
| 1,697 | 34,109 |
en
|
train
|
0.38.3
|
\section{The one-center $\delta$-interaction in one dimension }
In this subsection for convenience of the reader we establish initially a precise formulation for the point interaction determined by the formal linear differential operator
\begin{equation}\label{deltaop}
-\Delta_{\sigma}=-\frac{d^2}{dx^2}+\sigma \delta,
\end{equation}
defined on functions on the real line. The parameter $\sigma$ represents the coupling constant or strength attached to the point source located at $x=0$. We note that there are many approaches for studying the operator in (\ref{deltaop}), for instance, by the use of quadratic forms or by the self-adjoint extensions of symmetric operators. We also note that the quantum mechanics model in (\ref{deltaop}) has been studied into a more general framework when it is associated with the Kronig-Penney model in solid state physics (see Chapter III.2 in Albeverio {\it et al.} \cite{AGHH}) or when it is associated to singular rank one perturbations (Albeverio {\it et al.} \cite{ak}).
By following \cite{ak}, we consider the operator $A=-\frac{d^2}{dx^2}$ with domain $D(A)= H^2(\mathbb R)$ and the (closeable) symmetric restriction $A^0\equiv A|_{D(A^0)}$ with dense domain $D(A^0)=\{\psi\in D(A) : (\delta, \psi)\equiv\psi(0)=0\}$. Then we obtain that the {\it deficiency subspaces} of $A^0$,
\begin{equation}\label{sa1}
\mathcal D_+=\text{Ker}({A^0}^*-i),\quad{\rm{and}}\quad\mathcal D_-=\text{Ker}({A^0}^*+i),
\end{equation}
have dimension ({\it deficiency indexes}) equal to $1$. It is no difficult to see that these subspaces are generated, respectively, by $g_{+i}\equiv (A-i)^{-1}\delta$ and $g_{-i}\equiv (A+i)^{-1}\delta$, called {\it deficiency elements} and given explicitly by (see \cite{ak}),
\begin{equation}\label{def}
g_{\pm i}(x)=\frac{i}{2\sqrt{\pm i}}e^{i\sqrt{\pm i} |x|},\qquad Im \sqrt{\pm i} >0.
\end{equation}
We note that the Fourier transform of $g_{\pm i}$ are given by $\widehat{g_{\pm i}}(\xi)=\frac{1}{\xi^2\mp i}$
Next we present explicitly all the self-adjoint extensions of the symmetric operator $A^0$, which will be parameterized by the strength $\sigma$. By normalizing the deficiency elements $\widetilde{g}_{\pm i}=\frac{g_{\pm i}}{\|g_{\pm i\|}}$ and for convenience of notation we will continue to use $g_{\pm i}$, we have from the von Neumann's theory of self-adjoint extensions for symmetric operators (see \cite{RS}) that all the closed symmetric extensions of $A^0$ are self-adjoint and coincides with the restriction of the operator ${A^0}^*$. Moreover, for $\theta\in[0,2\pi)$ the self-adjoint extension $A^0(\theta)$ of $A^0$ is defined as follows:
\begin{equation}\label{sa4}
\left\{
\begin{aligned}
&D(A^0(\theta))=\{\psi+\lambda g_i+\lambda e^{i\theta}g_{-i} :\psi\in D(A^0), \lambda\in \mathbb C\},\\
&A^0(\theta)(\psi+\lambda g_i+\lambda e^{i\theta}g_{-i})={A^0}^*(\psi+\lambda g_i+\lambda e^{i\theta}g_{-i})=A^0\psi+i\lambda g_i-i\lambda e^{i\theta}g_{-i}.
\end{aligned}\right.
\end{equation}
Thus from \eqref{sa4} and \eqref{def} we obtain that for $\zeta\in D(A^0(\theta))$, in the form $\zeta=\psi+\lambda g_i+\lambda e^{i\theta}g_{-i}$, we have the basic expression
\begin{equation}\label{sa5}
\zeta'(0+)-\zeta'(0-)=-\lambda(1+e^{i\theta}).
\end{equation}
Next we find $\sigma$ such that $\sigma\zeta(0)=-\lambda(1+e^{i\theta})$. Indeed, $\sigma$ is given by the formula
\begin{equation}\label{sa6}
\sigma(\theta)=\frac{-2\cos(\theta/2)}{cos(\frac{\theta}{2}-\frac{\pi}{4})}.
\end{equation}
So, from now on we parameterize all self-adjoint extensions of $A^0$ with the help of $\sigma$. Thus we get:
\begin{theorem}\label{self} All self-adjoint extensions of $A^0$ are given for $-\infty<\sigma\leqq +\infty$ by
\begin{equation}\label{sa8}
\left\{
\begin{aligned}
-\Delta_{\sigma}&=-\frac{d^2}{dx^2}\\
D(-\Delta_{\sigma})&=\{\zeta\in H^1(\mathbb R)\cap H^2(\mathbb R-\{0\}): \zeta'(0+)-\zeta'(0-)=\sigma \zeta(0)\}.
\end{aligned}\right.
\end{equation}
The special case $\sigma=0$ just leads to the operator $-\Delta$ in $L^2(\mathbb R)$,
\begin{equation}\label{sa9}
-\Delta=-\frac{d^2}{dx^2},\qquad D(-\Delta)= H^2(\mathbb R),
\end{equation}
whereas the case $\sigma=+\infty$ yields a Dirichlet boundary condition at zero,
\begin{equation}\label{sa9a}
D(-\Delta_{+\infty})=\{\zeta\in H^1(\mathbb R)\cap H^2(\mathbb R-\{0\}): \zeta(0)=0\}.
\end{equation}
\end{theorem}
\textbf{Proof.} By the arguments sketched above we obtain easily that $A^0(\theta)\subset -\Delta_{\sigma}$,
with $\sigma=\sigma(\theta)$ given in \eqref{sa6}. But $-\Delta_{\sigma}$ is symmetric in the corresponding domain $D(-\Delta_{\sigma})$ for all $-\infty<\sigma\leqq +\infty$, which implies the relation $A^0(\theta)\subset -\Delta_{\sigma}\subset (-\Delta_{\sigma})^* \subset A^0(\theta)$.
It completes the proof of the Theorem.
\fin
Next, we recall the basic spectral properties of $-\Delta_\sigma$ which will be relevant for our results (see \cite{AGHH}).
\begin{theorem}\label{resol5b} Let $-\infty<\sigma\leqq +\infty$. Then the essential spectrum of $-\Delta_\sigma$ is the nonnegative real axis, $\Sigma_{ess}(-\Delta_\sigma)=[0,+\infty)$.
If $-\infty<\sigma<0$, $-\Delta_\sigma$ has exactly one negative, simple eigenvalue, i.e., its discrete spectrum $\Sigma_{dis}(-\Delta_\sigma)$ is $\Sigma_{dis}(-\Delta_\sigma)=\{{-\sigma^2/4}\}$, with a strictly (normalized) eigenfunction
$$
\Psi_\sigma(x)=\sqrt{\frac{-\sigma}{2}}e^{\frac{\sigma}{2}|x|}.
$$
If $\sigma\geqq 0$ or $\sigma=+\infty$, $-\Delta_\sigma$ has not discrete spectrum, $\Sigma_{dis}(-\Delta_\sigma)=\emptyset$.
\end{theorem}
\section{Two symmetric $\delta$-interaction in one dimension }
The one-dimensional Schr\"odinger operator with two symmetric delta interactions of strength $\alpha$ and placed at the point $\pm a$ is given formally by the linear differential operator
\begin{equation}\label{deltaop1}
-\Delta_{\alpha}=-\frac{d^2}{dx^2}+\alpha( \delta(\cdot-a)+\delta(\cdot+a)),
\end{equation}
defined on functions on the real line. By using the same notations as in last section, the symmetric operator $A^1=A|_{D(A^1)}$ with dense domain
$$
D(A^1)=\{\psi\in H^2(\mathbb R): \psi(\pm a)=0\},
$$
has deficiency indices (2, 2), and so from the Von Neumann-Krein theory we have that all self-adjoint extensions of $A^1$ are given by a four-parameter family of self-adjoint operators. Here we restrict to the case of so-called separated boundary conditions at each point $\pm a$. More specifically, we have the following theorem (see \cite{ABD}).
\begin{theorem}\label{self3} There is a family of self-adjoint extensions of $A^1$ given for $-\infty<\alpha\leqq +\infty$ by
\begin{equation}\label{sa8}
\left\{
\begin{aligned}
-\Delta_{\alpha}&=-\frac{d^2}{dx^2}\\
D(-\Delta_{\alpha})&=\{\zeta\in H^1(\mathbb R)\cap H^2(\mathbb R-\{\pm a\}): \zeta'(\pm a+)-\zeta'(\pm a-)=\alpha \zeta(\pm a)\},
\end{aligned}\right.
\end{equation}
The special case $\alpha=0$ just leads to the operator $-\Delta$ in $L^2(\mathbb R)$,
\begin{equation}\label{sa9}
-\Delta=-\frac{d^2}{dx^2},\qquad D(-\Delta)= H^2(\mathbb R),
\end{equation}
whereas the case $\alpha=+\infty$ yields a Dirichlet boundary condition at the point $\pm a$,
\begin{equation}\label{sa9a}
D(-\Delta_{+\infty})=\{\zeta\in H^1(\mathbb R)\cap H^2(\mathbb R-\{\pm a\}): \zeta( \pm a+)=\zeta( \pm a-)=0\}.
\end{equation}
\end{theorem}
Next, we establish the basic spectral properties of $-\Delta_\alpha$ which will be relevant for our results (see \cite{AGHH}).
\begin{theorem}\label{resol2de} Let $-\infty<\alpha\leqq +\infty$. Then the essential spectrum of $-\Delta_\alpha$ is the nonnegative real axis, $\Sigma_{ess}(-\Delta_\alpha)=[0,+\infty)$.
I) If $-\infty<\alpha<0$, then the discrete spectrum of $-\Delta_\alpha$, $\Sigma_{dis}(-\Delta_\alpha)$, consists of negative eigenvalues $\gamma$ given by the implicit equation
$$
(-2i \eta +\alpha)^2=\alpha^2e^{4i\eta a},\quad \eta=\sqrt{\gamma},\;\; Im \;\eta>0.
$$
Moreover, we have that:
\begin{enumerate}
\item[1)] if $a\leqq -\frac{1}{\alpha}$, then $\Sigma_{dis}(-\Delta_\alpha)=\{\gamma_1(a,\alpha)\}$, where $\gamma_1(a,\alpha)$ is defined by
$$
\gamma_1(a,\alpha)=-\frac{1}{4a^2}[W(-a\alpha e^{a\alpha})-a\alpha]^2,
$$
where $W(\cdot)$ is the Lambert special function (or product logarithm) defined by the equation $W(x)e^{W(x)}=x$.
\item[2)] if $a> -\frac{1}{\alpha}$, then $\Sigma_{dis}(-\Delta_\alpha)=\{\gamma_1(a,\alpha), \gamma_2(a,\alpha)\}$, where $\gamma_2(a,\alpha)$ is defined by
$$
\gamma_2(a,\alpha)=-\frac{1}{4a^2}[W(a\alpha e^{a\alpha})-a\alpha]^2.$$
\end{enumerate}
II) If $\alpha\geqq 0$ or $\alpha=+\infty$, $-\Delta_\alpha$ has not discrete spectrum, $\Sigma_{dis}(-\Delta_\alpha)=\emptyset$.
\end{theorem}
| 2,951 | 34,109 |
en
|
train
|
0.38.4
|
\section{The $\delta'$-interaction in one dimension }
In this subsection for convenience of the reader we establish a precise formulation for the point interaction determined by the formal linear differential operator
\begin{equation}\label{deltaop1}
-\Delta_{\beta}=-\frac{d^2}{dx^2}+\beta \delta',
\end{equation}
defined on functions on the real line. The parameter $\beta$ represents the coupling constant or strength attached to the point source located at $x=0$ and $ \delta'$ is the derivative of the $\delta$. By following Albeverio {\it et al.} \cite{AGHH}-\cite{ak}, the elements in the domain of the operator $-\Delta_{\beta}$ are characterized by suitable bilateral singular boundary conditions at the singularity (see (\ref{bc1})), while the real true action coincides with the laplacian out the singularity. At variance with the $\delta$ interaction, whose domain is contained in $H^1(\mathbb R)\cap H^2(\mathbb R-\{0\})$ (in particular in a continuous function set, see (\ref{sa8})), the latter has a domain contained only in $H^2(\mathbb R-\{0\})$ and so by allowing discontinuities of the elements at the position of the defect. More precisely, for $A^2=A|_{D(A^2)}$ being considered with dense domain
$$
D(A^2)=\{\psi\in H^2(\mathbb R): \psi(0)=\psi'(0)=0\}.
$$
$A^2$ has deficiency indices $(2,2)$ and hence it has a four-parameter family of self-adjoint. We are interested in the following one-parameter family of self-adjoint extensions (see \cite{AGHH}-\cite{ak}).
\begin{theorem}\label{selfd} There is a family of self-adjoint extensions of $A^2$ given for $-\infty<\beta\leqq +\infty$ by
\begin{equation}\label{sa8d}
\left\{
\begin{aligned}
-\Delta_{\beta}&=-\frac{d^2}{dx^2}\\
D(-\Delta_{\beta})&=\{\zeta\in H^2(\mathbb R-\{0\}): \zeta'(0+)=\zeta'(0-), \zeta(0+)-\zeta(0-)=\beta\zeta'(0-)\},
\end{aligned}\right.
\end{equation}
The special case $\beta=0$ just leads to the operator $-\Delta$ in $L^2(\mathbb R)$,
\begin{equation}\label{sa9d}
-\Delta=-\frac{d^2}{dx^2},\qquad D(-\Delta)= H^2(\mathbb R),
\end{equation}
whereas the case $\beta=+\infty$ yields a Neumann boundary condition at zero and decouples $(-\infty, 0)$ and $(0,+\infty)$, i.e.,
\begin{equation}\label{sa9d}
D(-\Delta_{+\infty})=\{\zeta\in H^2(\mathbb R-\{0\}): \zeta'(0+)=\zeta'(0-)=0\}.
\end{equation}
\end{theorem}
Note that the functions in the domain of $\delta'$ have a jump at the origin, and the left and right derivatives coincide. Next, we establish the basic spectral properties of $-\Delta_\beta$ which will be relevant for our results (see \cite{AGHH}).
\begin{theorem}\label{resol5d} Let $-\infty<\beta\leqq +\infty$. Then the essential spectrum of $-\Delta_\beta$ is the nonnegative real axis, $\Sigma_{ess}(-\Delta_\beta)=[0,+\infty)$.
If $-\infty<\beta<0$, $-\Delta_\beta$ has exactly one negative simple eigenvalue, i.e., its discrete spectrum $\Sigma_{dis}(-\Delta_\beta)$ is $\Sigma_{dis}(-\Delta_\beta)=\{-\frac{4}{\beta^2}\}$, with a (normalized) eigenfunction
$$
\Phi_\beta(x)=\Big(-\frac{2}{\beta}\Big)^{\frac12}\text{sign}(x) e^{\frac{2}{\beta}|x|}.
$$
If $\beta\geqq 0$ or $\beta=+\infty$, $-\Delta_\beta$ has not discrete spectrum, $\Sigma_{dis}(-\Delta_\beta)=\emptyset$.
\end{theorem}
| 1,031 | 34,109 |
en
|
train
|
0.38.5
|
\subsection{The linear propagator }
\subsubsection{The case $\mu(x)=\sigma \delta$.}
Next we determine the linear propagator $G_\sigma=e^{i(\Delta+\sigma\delta)t}$(unitary group) determined by the linear system associated with (\ref{SCH0}),
\begin{equation}
\left \{
\begin{aligned}
i\partial_{t}u&=-(\Delta u+\sigma\delta)u\equiv H_\sigma u, \label{SCH1}\\
u(0) & =u_{0},
\end{aligned} \right.
\end{equation}
where we are using the notation $H_\sigma=-\Delta_{-\sigma}$.
We will use the representation of the propagator in terms of the eigenfunctions (associated to discrete eigenvalues) and generalized eigenfunctions (see Iorio \cite{Io}, Holmer {\it et al.} \cite{Holmer5} and Duch\^ene {\it et al.} \cite{DMW}). Indeed, the family of generalized eigenfunctions $\{\psi_\lambda\}_{\lambda\in \mathbb R}$ will be such that satisfy
\begin{equation}
\left\{
\begin{aligned}\label{eigengen}
&H_\sigma \psi_\lambda=\lambda^2 \psi_\lambda,\qquad \psi_\lambda\;\text{continuous and}\\\
& \psi'_\lambda(0+)-\psi'_\lambda(0-)=\sigma\psi_\lambda(0).
\end{aligned}\right.
\end{equation}
Hence we obtain the following family of special solutions, $e_{\pm}(x,\lambda)$ to (\ref{eigengen}), as follows
\begin{equation}\label{efun}
e_{\pm}(x,\lambda)=t_\sigma(\lambda)e^{\pm i\lambda x}\chi^0_{\pm}+(e^{\pm i\lambda x}+r_\sigma(\lambda)e^{\mp i\lambda x})\chi^0_{\mp},
\end{equation}
where $\chi^0_{+}$ is the characteristic function of $[0,+\infty)$ and $\chi^0_{-}$ is the characteristic function of $(-\infty, 0]$. $t_\sigma$ and $r_\sigma$ are the transmission and reflection coefficients:
\begin{equation}\label{tr}
t_\sigma(\lambda)=\frac{2i\lambda}{2i\lambda-\sigma},\quad r_\sigma(\lambda)=\frac{\sigma}{2i\lambda-\sigma}.
\end{equation}
They satisfy the following two equations:
\begin{equation}\label{tr2}
| t_\sigma(\lambda)|^2+| r_\sigma(\lambda)|^2=1,\quad r_\sigma(\lambda)+1=t_\sigma(\lambda).
\end{equation}
Next, by defining the family $\{\psi_\lambda\}_{\lambda\in \mathbb R}$ as
$$
\psi_\lambda (x)=\left\{
\begin{aligned}
& e_+(x,\lambda)\;\; \;\;for \;\; \lambda\geqq 0\\
&e_-(x, -\lambda)\;\; for \;\; \lambda < 0
\end{aligned} \right.
$$
we obtain from Theorem \ref{resol5b} the following relations (see \cite{DMW}, \cite{Io});
\begin{enumerate}\label{orthorela}
\item[1)] $\int_{\mathbb R} \Psi_\sigma(x) \overline{\psi_\lambda(x)}dx=0$, \;\; for all $\lambda \in \mathbb R$ and $ \sigma <0$,\\
\item[2)] $ \int_{\mathbb R} \psi_\mu(x) \overline{\psi_\lambda (x)}dx=\delta (\lambda-\mu)$, \;\; for all $\mu, \lambda \in\mathbb R$,\\
\item[3)] $\Psi_\sigma(x)\Psi_\sigma(y)+ \int_{\mathbb R} \psi_\lambda(x) \overline{\psi_\lambda (y)}d\lambda=\delta (x-y)$, \;\; $ \lambda \in \mathbb R$, $\sigma <0$.
\end{enumerate}
We recall that the relation 3) above, called the {\it completeness relations}, in the case $\sigma >0$ is reads as $\int_{\mathbb R} \psi_\lambda(x) \overline{\psi_\lambda}(y)d\lambda=\delta (x-y)$ (the proof of 3) for the family $\{\psi_\lambda\}_{\lambda\in \mathbb R}$ can be showed by following the ideas in the proof of Theorem \ref{expli} below). Moreover, the family $\{\psi_\lambda\}_{\lambda\in \mathbb R}$ allows us to define the {\it generalized Fourier transform}
\begin{equation}
\mathcal F (f)(\lambda)=\int_{\mathbb R} f(x)\overline{\psi_\lambda (x)}dx,
\end{equation}
and its formal adjoint $\mathcal G (g)(x)=\int_{\mathbb R} \psi_\lambda (x) g(\lambda)d\lambda$. Hence, from 2) we obtain immediately that $\mathcal G$ is the inverse Fourier transform, namely,
$$
f(\lambda)=f\ast \delta (\lambda)=\int_{\mathbb R} \overline{\psi_\lambda (x)}\int_{\mathbb R} f(\mu)\psi_\mu (x)d\mu dx=\mathcal F(\mathcal G g)(\lambda).
$$
Moreover, from the completeness relations 3) we obtain for every $f\in L^2(\mathbb R)$ the following (orthogonal) expansion in eigenfunctions of $H_\sigma$,
\begin{equation}
f= \langle f, \Phi_\sigma\rangle \Phi_\sigma + \int_{\mathbb R} \mathcal F( f)(\lambda)\psi_\lambda(x)d\lambda.
\end{equation}
Thus for $u\in C(\mathbb R; L^2(\mathbb R))$ being a solution of (\ref{SCH1}), the method of separation of variables implies that
\begin{equation}\label{solu1}
u(x,t)=e^{-it H_\sigma} u_0(x)=
e^{i\frac{\sigma^2}{4}t} \langle u_0, \Phi_\sigma\rangle \Phi_\sigma (x) +\int_{\mathbb R}e^{-i\lambda^2 t}\mathcal F (u_0)(\lambda)\psi_\lambda (x)d\lambda.
\end{equation}
In the next Theorem we describe explicitly the propagator $e^{-it H_\sigma}$ in terms of the free propagator of the Schr\"odinger equation $e^{it\Delta }$ (see Holmer {\it et al.} \cite{Holmer5}, Datchev\&Holmer \cite{Holmer3}). In the Appendix we present a different proof based in the fundamental solution associated to (\ref{SCH1}).
\begin{theorem} \label{expli} Suppose that $ \phi\in L^1(\mathbb R)$ with $supp\; \phi \subset (-\infty, 0]$. Then,
\begin{enumerate}
\item[1)] Para $\sigma \geqq 0$, we have
\begin{equation}\label{pospro}
e^{-it H_\sigma} \phi(x)= e^{it \Delta} (\phi\ast \tau_\sigma) (x) \chi^0_{+} + \Big[e^{it \Delta} \phi(x) + e^{it \Delta} (\phi\ast \rho_\sigma) (-x) \Big ]\chi^0_{-}
\end{equation}
where
$$
\rho_\sigma(x)=-\frac{\sigma}{2} e^{\frac{\sigma}{2} x}\chi^0_{-},\;\; \tau_\sigma(x)=\delta (x)+ \rho_\sigma(x).
$$
\item[ 2)] Para $\sigma < 0$, we have
\begin{equation}\label{negpro}
e^{-it H_\sigma} \phi(x)= e^{i\frac{\sigma^2}{4} t} P\phi (x)+ e^{it \Delta} (\phi\ast \tau_\sigma) (x) \chi^0_{+} + \Big[e^{it \Delta} \phi(x) + e^{it \Delta} (\phi\ast \rho_\sigma) (-x) \Big ]\chi^0_{-}
\end{equation}
where
$$
\rho_\sigma(x)=\frac{\sigma}{2} e^{\frac{\sigma}{2} x}\chi^0_{+},\;\; \tau_\sigma(x)=\delta (x)+ \rho_\sigma(x),
$$
and $P$ is the $L^2(\mathbb R)$-orthogonal projection onto the eigenfunction $\Phi_\sigma$, $P\phi=\langle \phi, \Phi_\sigma\rangle \Phi_\sigma $.
\end{enumerate}
\end{theorem}
\begin{remark} We observe the following:
\begin{itemize}
\item[1) ] The Fourier transform de $\rho_\sigma$ for every sign of $\sigma$ is given by $\widehat{\rho_\sigma}(\lambda)=r_\sigma(\lambda)$ and so $\widehat{\tau_\sigma}(\lambda)=1+r_\sigma(\lambda)=t_\sigma(\lambda)$.
\item[2) ] Formula (\ref{pospro}) and (\ref{negpro}) will allow us to estimate the operator norm of $e^{-it H_\sigma}$ using $e^{it \Delta}$.
\end{itemize}
\end{remark}
\textbf{Proof.} We only consider the case $\sigma\geqq 0$. From (\ref{solu1}), without the term of projection, we have from the definition of the family $\{\psi_\lambda\}$ and a change of variable that
\begin{equation}\label{grupo1}
e^{-it H_\sigma} \phi(x)=\int_{\mathbb R}\phi(y)\int_{0}^\infty e^{-it\lambda^2}\Big (e_+(x,\lambda)\overline{e_+(y,\lambda)}+e_-(x,\lambda)\overline{e_-(y,\lambda)}\Big)d\lambda dy.
\end{equation}
Next, we compute first
\begin{equation}
\begin{aligned}
& \int_{\mathbb R}\phi(y)\overline{e_+(y,\lambda)}dy=\int_{-\infty}^0 \phi(y)e^{-i\lambda y}dy +\overline{r_\sigma(\lambda)}\int_{-\infty}^0 \phi(y)e^{i\lambda y}dy=\widehat{\phi}(\lambda)+r_\sigma(-\lambda)\widehat{\phi}(-\lambda),\\
&\qquad\qquad\qquad \qquad\int_{\mathbb R}\phi(y)\overline{e_-(y,\lambda)}dy=t_\sigma(-\lambda)\widehat{\phi}(-\lambda),
\end{aligned}
\end{equation}
so for $x>0$ we have from (\ref{grupo1}) and the fact $r_\sigma(-\lambda)t_\sigma(\lambda)+r_\sigma(\lambda)t_\sigma(-\lambda)=0$ that
\begin{equation}
e^{-it H_\sigma} \phi(x)=\int_{\mathbb R}e^{-it\lambda^2}t_\sigma(\lambda)\widehat{\phi}(\lambda)e^{i\lambda x}d\lambda=e^{it \Delta}(\tau_\sigma\ast\phi)(x),
\end{equation}
where $\widehat{\tau_\sigma}(\lambda)=t_\sigma(\lambda)$. Similarly, since $r_\sigma(-\lambda)r_\sigma(\lambda)+t_\sigma(-\lambda)t_\sigma(\lambda)=1$, we have for $x<0$
\begin{equation}
e^{-it H_\sigma} \phi(x)=\int_{\mathbb R}e^{-it\lambda^2}(\widehat{\phi}(\lambda)e^{i\lambda x}+r_\sigma(\lambda)\widehat{\phi}(\lambda)e^{-i\lambda x})d\lambda=e^{it \Delta}\phi(x)+e^{it \Delta}(\rho_\sigma\ast\phi)(-x),
\end{equation}
where $\widehat{\rho_\sigma}(\lambda)=r_\sigma(\lambda)$.
\fin
\subsubsection{The case $\mu(x)=\alpha (\delta(\cdot-a)+\delta(\cdot+a))$.}
Next we determine the linear propagator $M_\alpha(t)=e^{-itU_\alpha} $ (unitary group) determined by the linear system associated with (\ref{SCH0}),
\begin{equation}\label{beta3}
\left \{
\begin{aligned}
i\partial_{t}u&=-(\Delta u+\alpha (\delta(\cdot-a)+\delta(\cdot+a))u\equiv U_\alpha u, \\
u(0) & =u_{0},
\end{aligned} \right.
\end{equation}
where we are using the notation $U_\alpha=-\Delta_{-\alpha}$.
We will use the fundamental solution $F_\alpha(x,y;t)$ to the Schr\"odinger equation (\ref{beta}) for obtaining the propagator (unitary group). Then we have the representation
\begin{equation}\label{uni}
e^{-itU_\alpha}f(x)=\int_{\mathbb R} F_\alpha(x,y;t)f(y)dy.
\end{equation}
Indeed, from \cite{ABD} and \cite{KS} we have for $S(x;t)$ denoting the free propagator in $\mathbb R$, i.e.
\begin{equation}\label{free}
S(x,t)= \frac{e^{-x^2/{4it}}}{(4i\pi t)^{1/2}},\quad t>0
\end{equation}
and so $e^{it\Delta}f(x)=S(x;t)\ast_x f(x)$, the following expression for $a\alpha \neq -1$:
\begin{enumerate}
\item[1)] For $\alpha>0$
\begin{equation}\label{ker51}
F_\alpha(x,y;t)=S(x-y;t)- \frac{1}{2\pi i}\int_{\mathbb R}e^{-i\xi ^2t}\frac{f_\alpha(x,y;\xi)}{(2\xi+i\alpha)^2+\alpha^2e^{i4\xi a}}d\xi
\end{equation}
with $f_\alpha(x,y;\xi)=\sum_{j=1}^4 L_\alpha^j(x,y;\xi)$ and
\begin{align}
L_\alpha^1(x,y;\xi)&=-\alpha(2\xi+i\alpha)e^{i\xi|x+a|}e^{i\xi|y+a|},\qquad L_\alpha^4(x,y;\xi)=L_\alpha^1(-x,-y;\xi)\\
L_\alpha^2(x,y;\xi)&=i\alpha^2e^{2i\xi a}e^{i\xi|x+a|}e^{i\xi|y-a|},\qquad\;\;\;\;\;\;\; L_\alpha^3(x,y;\xi)=L_\alpha^2(-x,-y;\xi).
\end{align}
\item[2)] For $\alpha<0$,
\begin{equation}\label{ker6}
F_\alpha(x,y;t)=e^{-it\gamma_1} \Gamma_1(x) \Gamma_1(y)+e^{-it\gamma_2} \Gamma_2(x) \Gamma_2(y) +F_{-\alpha}(x,y;t)
\end{equation}
where $ \Gamma_1$ and $ \Gamma_2$ are the normalized eigenfunction associated with the eigenvalues $\gamma_1$ and $\gamma_2$.
\end{enumerate}
\begin{remark} We observe the following:
\begin{itemize}
\item[1) ] The case $a\alpha=-1$ is assumed for technical reasons (see \cite{KS})
\item[2) ] For $a\alpha=-1$ we obtain that only $\gamma_1$ remains as an eigenvalue in the discrete spectrum.
\end{itemize}
\end{remark}
| 3,814 | 34,109 |
en
|
train
|
0.38.6
|
\subsubsection{The case $\mu(x)=\alpha (\delta(\cdot-a)+\delta(\cdot+a))$.}
Next we determine the linear propagator $M_\alpha(t)=e^{-itU_\alpha} $ (unitary group) determined by the linear system associated with (\ref{SCH0}),
\begin{equation}\label{beta3}
\left \{
\begin{aligned}
i\partial_{t}u&=-(\Delta u+\alpha (\delta(\cdot-a)+\delta(\cdot+a))u\equiv U_\alpha u, \\
u(0) & =u_{0},
\end{aligned} \right.
\end{equation}
where we are using the notation $U_\alpha=-\Delta_{-\alpha}$.
We will use the fundamental solution $F_\alpha(x,y;t)$ to the Schr\"odinger equation (\ref{beta}) for obtaining the propagator (unitary group). Then we have the representation
\begin{equation}\label{uni}
e^{-itU_\alpha}f(x)=\int_{\mathbb R} F_\alpha(x,y;t)f(y)dy.
\end{equation}
Indeed, from \cite{ABD} and \cite{KS} we have for $S(x;t)$ denoting the free propagator in $\mathbb R$, i.e.
\begin{equation}\label{free}
S(x,t)= \frac{e^{-x^2/{4it}}}{(4i\pi t)^{1/2}},\quad t>0
\end{equation}
and so $e^{it\Delta}f(x)=S(x;t)\ast_x f(x)$, the following expression for $a\alpha \neq -1$:
\begin{enumerate}
\item[1)] For $\alpha>0$
\begin{equation}\label{ker51}
F_\alpha(x,y;t)=S(x-y;t)- \frac{1}{2\pi i}\int_{\mathbb R}e^{-i\xi ^2t}\frac{f_\alpha(x,y;\xi)}{(2\xi+i\alpha)^2+\alpha^2e^{i4\xi a}}d\xi
\end{equation}
with $f_\alpha(x,y;\xi)=\sum_{j=1}^4 L_\alpha^j(x,y;\xi)$ and
\begin{align}
L_\alpha^1(x,y;\xi)&=-\alpha(2\xi+i\alpha)e^{i\xi|x+a|}e^{i\xi|y+a|},\qquad L_\alpha^4(x,y;\xi)=L_\alpha^1(-x,-y;\xi)\\
L_\alpha^2(x,y;\xi)&=i\alpha^2e^{2i\xi a}e^{i\xi|x+a|}e^{i\xi|y-a|},\qquad\;\;\;\;\;\;\; L_\alpha^3(x,y;\xi)=L_\alpha^2(-x,-y;\xi).
\end{align}
\item[2)] For $\alpha<0$,
\begin{equation}\label{ker6}
F_\alpha(x,y;t)=e^{-it\gamma_1} \Gamma_1(x) \Gamma_1(y)+e^{-it\gamma_2} \Gamma_2(x) \Gamma_2(y) +F_{-\alpha}(x,y;t)
\end{equation}
where $ \Gamma_1$ and $ \Gamma_2$ are the normalized eigenfunction associated with the eigenvalues $\gamma_1$ and $\gamma_2$.
\end{enumerate}
\begin{remark} We observe the following:
\begin{itemize}
\item[1) ] The case $a\alpha=-1$ is assumed for technical reasons (see \cite{KS})
\item[2) ] For $a\alpha=-1$ we obtain that only $\gamma_1$ remains as an eigenvalue in the discrete spectrum.
\end{itemize}
\end{remark}
\subsubsection{The case $\mu(x)=\beta \delta'$.}
Next we determine the linear propagator $J_\beta(t)=e^{i(\Delta+\beta\delta')t}$ (unitary group) determined by the linear system associated with (\ref{SCH0}),
\begin{equation}\label{beta}
\left \{
\begin{aligned}
i\partial_{t}u&=-(\Delta u+\beta\delta')u\equiv K_\beta u, \\
u(0) & =u_{0},
\end{aligned} \right.
\end{equation}
where we are using the notation $K_\beta=-\Delta_{-\beta}$.
We will use the fundamental solution $S_\beta(x,y;t)$ to the Schr\"odinger equation (\ref{beta}) for obtaining the propagator (unitary group). Then we have the representation
\begin{equation}\label{uni}
e^{-itK_\beta}f(x)=\int_{\mathbb R} S_\beta(x,y;t)f(y)dy.
\end{equation}
Indeed, from \cite{AGHH} we have for $S(x;t)$ denoting the free propagator in (\ref{free}) the following:
\begin{enumerate}
\item[1)] For $\beta>0$
\begin{equation}\label{ker5}
S_\beta(x,y;t)=S(x-y;t)+\text{sgn}(xy) S(|x|+|y|;t)+ \frac{2}{\beta}\int_0^\infty \text{sgn}(xy)e^{-\frac{2}{\beta} s} S(s+|x|+|y|;t)ds
\end{equation}
\item[2)] For $\beta<0$,
\begin{equation}\label{ker6}
\begin{aligned}
S_\beta(x,y;t)=&S(x-y;t)+\text{sgn}(xy) S(|x|+|y|;t)+ e^{i\frac{4}{\beta^2} t} \Phi_\beta(x) \Phi_\beta(y)\\
&-\frac{2}{\beta} \int_0^\infty \text{sgn}(xy)e^{\frac{2}{\beta} s} S(s-|x|-|y|;t)ds
\end{aligned}
\end{equation}
where $ \Phi_\beta$ is defined in Theorem \ref{resol5d}.
\end{enumerate}
\subsubsection{Dispersive Estimates}
The following proposition extends the well known estimates for the free propagator $e^{it\Delta}$,
\begin{equation}\label{stric0}
\left\Vert e^{it\Delta}f(t)\right\Vert _{p^{\prime}}\leq C_0t^{-\frac{1}{2}
(\frac{2}{p}-1)}\left\Vert f\right\Vert _{p},
\end{equation}
to the the groups $G_\sigma(t)=e^{-it H_\sigma}$, $M_\alpha(t)=e^{-it U_\alpha}$ and $J_\beta(t)=e^{i(\Delta+\beta\delta')t} $ in the one-dimensional case. We denote by $W_1$ the group $G_\sigma$, $W_2$ the group $M_\alpha$ and by $W_3$ the group $J_\beta$ .
\begin{proposition} \label{strich}
Let $p\in [1, 2]$ and $p'$ be such that $\frac{1}{p}+\frac{1}{p'}=1$. Then we have:
Suppose that $u(x,t)=W_i(t)f(x)$, $i=1,2,3$, is the solution of the linear equation (\ref{SCH1}), (\ref{beta3}) and (\ref{beta}), respectively. Then:
\begin{itemize}
\item[1)] for $\sigma\geqq 0$, $\alpha \geqq 0$ and $\beta\geqq 0$,
\begin{equation}\label{stric2}
\left\Vert W_i(t)f\right\Vert _{p^{\prime}}\leq C|t|^{-\frac{1}{2}
(\frac{2}{p}-1)}\left\Vert f\right\Vert _{p},
\end{equation}
where $C>0$ is independent of $f$ and $t\neq0$.
\item[2)] for $\sigma< 0$, $a\leqq -\frac{1}{\alpha}$ ($\alpha<0$) and $\beta<0$,
\begin{equation}\label{stric3}
\left\Vert W_i(t)f-e^{i\alpha_i t} P_if\right\Vert _{p^{\prime}}\leq C|t|^{-\frac{1}{2}
(\frac{2}{p}-1)}\left\Vert f\right\Vert _{p},
\end{equation}
where $\alpha_1=\frac{\sigma^2}{4}$, $P_1f= \langle f, \Psi_\sigma\rangle \Psi_\sigma$, $\alpha_2=-\gamma_1(a,\alpha)$, $P_2f= \langle f, \Gamma_1\rangle \Gamma_1$ and $\alpha_3=\frac{4}{\beta^2}$, $P_3f= \langle f, \Phi_\beta\rangle \Phi_\beta$, and $C>0$ is independent of $f$ and $t\neq0$.
\item[3)] for $\alpha<0$, $a> -\frac{1}{\alpha}$,
\begin{equation}\label{stric4}
\left\Vert W_2(t)f-\sum_{j=1}^2e^{-i\gamma_j t} Q_jf\right\Vert _{p^{\prime}}\leq C|t|^{-\frac{1}{2}
(\frac{2}{p}-1)}\left\Vert f\right\Vert _{p},
\end{equation}
where $\gamma_j=\gamma_j(a,\alpha)$, $j=1,2$,
$Q_1f= \langle f, \Gamma_1\rangle \Gamma_1$, $Q_2f= \langle f, \Gamma_2\rangle \Gamma_2$, and $C>0$ is independent of $f$ and $t\neq0$.
\end{itemize}
\end{proposition}
\textbf{Proof.} i) We consider $\sigma>0$. Initially $G_{\sigma}(t)$ is a unitary group on $L^2(\Bbb R)$, $\|G_{\sigma}\phi(t)\|_2=\|\phi\|_2$ for all $t\in \mathbb R$. Let $\phi\in L^1(\mathbb R)$ and $R\phi(x)=\phi(-x)$. Then for $\phi^-=\phi \chi^0_-$ and $\phi^+=\phi R\chi^0_+$ we have the decomposition $\phi=\phi^-+R\phi^+$. Hence since $supp\; \phi^+\subset (-\infty, 0]$, $RG_{\sigma}=G_{\sigma}R$ and $R(f\ast Rg)=(Rf)\ast g$
we obtain from the following equality,
\begin{align}
G_{\sigma}\phi(t)&=[e^{it\Delta}\phi^-+e^{it\Delta}(\phi^-\ast \rho_\sigma)]\chi^0_-+e^{it\Delta}(\phi^-\ast \tau_\sigma)\chi^0_+\\
&+[e^{it\Delta}R\phi^+ +e^{it\Delta}(\phi^+\ast \rho_\sigma)]\chi^0_ + +e^{it\Delta}(R\phi^+\ast R\tau_\sigma)\chi^0_-.
\end{align}
Therefore from (\ref{stric0}) and applying Young's inequality we obtain for $t\neq 0$
\begin{align}
\|G_{\sigma}\phi(t)\|_\infty&\leqq \frac{C_0}{\sqrt{|t|}}(\|\phi^-\|_1+\|\phi^-\ast \rho_\sigma\|_1+\|\phi^-\ast \tau_\sigma\|_1+\|R\phi^+\|_1+\|\phi^+\ast \rho_\sigma\|_1 +\|R\phi^+\ast R\tau_\sigma\|_1)\\
& \leqq \frac{C_0}{\sqrt{|t|}}(3+4\|\rho_\sigma\|_1)\|\phi\|_1=\frac{C}{\sqrt{|t|}}\|\phi\|_1
\end{align}
where $C=C(\sigma)$. By the Riesz-Thorin interpolation theorem we obtain (\ref{stric2}). The case $\sigma<0$ follows similarly from the expression (\ref{negpro}).
ii) Let $\beta>0$. From (\ref{ker5}) we obtain immediate for $x,y\in \mathbb R$ that
$$
|S_\beta(x,y,t)|\leqq C|t|^{-\frac12}.
$$
So, from (\ref{uni}) we have $\|J_\beta(t)\phi\|_\infty\leqq \frac{C}{\sqrt{|t|}}\|\phi\|_1$. By following a similar analysis as in the later case we obtain (\ref{stric2}).
iii) From the representations in (\ref{negpro}) and (\ref{ker6}) we get immediately (\ref{stric3}).
iv) The case of the group $W_2$ for $a\alpha \neq -1$, it follows of the estimate
$$
\|e^{-it U_\alpha}P_c f\|_\infty\leqq C t^{-1/2}\|f\|_1,
$$
for $t>0$ and $P_c$ being the spectral projector of $U_\alpha$ on its continuous spectrum.
\fin
| 3,288 | 34,109 |
en
|
train
|
0.38.7
|
\section{
Weak-$L^{p}$ Solutions}
In this section we focus our study of global solutions for the Cauchy problem
\begin{equation}
\left\{
\begin{aligned}
i\partial_{t}u+\Delta u+ \mu(x)u& =\lambda\left\vert u\right\vert
^{\rho-1}u,\ x\in\mathbb{R},\ t\in\mathbb{R},\label{SCHd}\\
u(x,0) & =u_{0},
\end{aligned}\right.
\end{equation}
for $\mu(x)=\sigma \delta$ in the spaces $L^{(p, \infty)}(\mathbb R)$, which are called weak-$L^p$ or Marcinkiewicz spaces. The cases $\mu(x)=\alpha(\delta(\cdot-a)+ \delta(\cdot+a))$ and $\mu(x)=\beta \delta'$ are treatment similarly.
We start by recalling some facts about the weak spaces $L^{(p, \infty)}(\mathbb R)$. For $1<r\leq
\infty,$ we recall that a measurable function $f$ defined on $\mathbb{R}$
belongs to $L^{(r,\infty)}(\mathbb{R})$ if the norm
\[
\Vert f\Vert_{(r,\infty)}=\sup_{t>0}t^{\frac{1}{r}}f^{\ast\ast}(t)
\]
is finite, where
\[
f^{\ast\ast}(t)=\frac{1}{t}\int_{0}^{t}f^{\ast}(s)\mbox{ }ds,
\]
and $ f^{\ast}$ is the {\it decreasing rearrangement} of $f$ with regard to the Lebesgue measure $\nu$, namely,
\[
f^{\ast}(t)=\inf\{s>0:\nu(\{x\in\mathbb{R}:|f(x)|>s\})\leq t\},\text{ }t>0,
\]
The space $L^{(r,\infty)}$ with the norm $\Vert f\Vert_{(r,\infty)}$ is a Banach space. We have the continuous inclusion $L^{r}(\mathbb{R})\subset$ $L^{(r,\infty)}(\mathbb{R})$. Moreover, the H\"{o}lder's inequality holds true in this framework, namely
\begin{equation}
\Vert fg\Vert_{(r,\infty)}\leq C\Vert f\Vert_{(q_{1},\infty)}\Vert
g\Vert_{(q_{2},\infty)}, \label{Holder1}
\end{equation}
for $1<q_{1},q_{2}<\infty$, $\frac{1}{q_{1}}+\frac{1}{q_{2}}<1$ and $\frac
{1}{r}=\frac{1}{q_{1}}+\frac{1}{q_{2}}$, where $C>0$ depends only on $r$. Lastly, we have the Lorentz spaces $L^{(p,q)}(\mathbb R)$ that can be constructed via real interpolation; indeed, $L^{(p,q)}(\mathbb R)=(L^{1}(\mathbb R),L^{\infty}(\mathbb R))_{1-\frac{1}{p},q}$, $1<p<\infty$. They have the interpolation property
\begin{equation}\label{inter}
(L^{(p_0, q_0)}(\mathbb R), L^{(p_1, q_1)}(\mathbb R))_{\theta, q}=L^{(p,q)}(\mathbb R),
\end{equation}
provided $0<p_0<p_1<\infty$, $0<\theta<1$, $\frac{1}{p}=\frac{1-\theta}{p_0}+\frac{\theta}{p_1}$, $1\leqq q_0, q_1, q\leqq \infty$, where $(\cdot,\cdot)_{\theta, q}$ stands for the real interpolation spaces constructed via the $K$-method. For
further details about weak-$L^{r}$ and Lorentz spaces see \cite{BL} and Grafakos \cite{Gra}.
From (\ref{inter}) we obtain our main estimate for the group $G_\sigma$ in Lorentz spaces. A similar result is obtained for the groups $M_\alpha$ and $J_\beta$.
\begin{lemma}\label{grupint1}
Let $1\leqq d\leqq \infty$, $p' \in (2,\infty)$, and $p\in (1,2)$. If $p'$ satisfies $\frac{1}{p}+\frac{1}{p'}=1$, then there exists a constant $C>0$ such that:
\begin{enumerate}
\item[1)] for $\sigma\geqq 0$,
\begin{equation}\label{grupint2}
\left\Vert G_{\sigma}(t)f\right\Vert _{(p^{\prime},d)}\leq C|t|^{-\frac
{1}{2}(\frac{2}{p}-1)}\left\Vert f\right\Vert _{(p, d)},
\end{equation}
for all $f\in L^{(p,d)}(\mathbb R)$ and all $t\neq 0$.
\item[2)] for $\sigma< 0$,
\begin{equation}\label{grupint3}
\left\Vert G_{\sigma}(t)f-e^{i\frac{\sigma^2}{4}t} P_1f\right\Vert _{(p^{\prime},d)}\leq C|t|^{-\frac{1}{2}(\frac{2}{p}-1)}\left\Vert f\right\Vert _{(p, d)},
\end{equation}
for all $f\in L^{(p,d)}(\mathbb R)$ and all $t\neq 0$.
\end{enumerate}
\end{lemma}
\textbf{Proof.} We only consider the case $\sigma\geqq 0$ because for $\sigma<0$ the analysis is similar. Let fixed $t\neq 0$ and let $1<p_0<p<p_1<2$. From the $L^p=L^{(p,p)}$ estimate of the Schr\" odinger group in Proposition \ref{strich}, we have that $G_\sigma (t):L^{p_0}\to L^{p_0'}$ and $G_\sigma (t):L^{p_1}\to L^{p_1'}$ satisfy
$$
\|G_\sigma (t)\|_{p_0\to p_0'}\leqq C|t|^{-\frac
{1}{2}(\frac{2}{p_0}-1)},\qquad \|G_\sigma (t)\|_{p_1\to p_1'}\leqq C|t|^{-\frac
{1}{2}(\frac{2}{p_1}-1)}
$$
with $\frac{1}{p_0}+\frac{1}{p_0'}=1$ and $ \frac{1}{p_1}+\frac{1}{p_1'}=1$. Hence, for $\lambda\in (0,1)$, $\frac{1}{p}=\frac{1-\lambda}{p_0}+\frac{\lambda}{p_1}$, and $\frac{1}{p' }=\frac{1-\lambda}{p'_0}+\frac{\lambda}{p'_1}$, we obtain from (\ref{inter}) that
$$
\|G_\sigma (t)\|_{(p,d) \to (p',d)}\leqq \|G_\sigma (t)\|_{p_0\to p_0'}^\lambda\|G_\sigma (t)\|_{p_1\to p_1'}^{(1-\lambda)}\leqq C |t|^{-\frac{1}{2}(\frac{2}{p}-1)},
$$
which gives (\ref{grupint2}).
\fin
Next we establish the main results of this section. From now on we focus in the case of $\mu(x)=\sigma\delta$ in (\ref{SCHd}), since for the case of the potential being the two symmetric deltas and the derivative of the Dirac- delta we have a similar analysis. We start by defining $\mathcal{L}_{\vartheta
}^{\infty}$ as the Banach space of all Bochner measurable functions
$u:\mathbb{R}\rightarrow$ $L^{(\rho+1,\infty)}$ endowed with the norm
\begin{equation}
\Vert u\Vert_{\mathcal{L}_{\vartheta }^{\infty}}=\sup_{-\infty<t<\infty
}|t|^{\vartheta }\Vert u(t)\Vert_{(\rho+1,\infty)}, \label{Norma1}
\end{equation}
where
\begin{equation}
\vartheta =\frac{1}{\rho-1}-\frac{1}{2(\rho+1)}. \label{exponent1}
\end{equation}
Let us also define the initial data space $\mathcal{E}_{0}$ as the set of all
$u\in\mathcal{S}^{\prime}(\mathbb{R})$ such that the norm
\[
\Vert u_{0}\Vert_{\mathcal{E}_{0}}=\sup_{-\infty<t<\infty}|t|^{\vartheta }\Vert
G_{\sigma}(t)u_{0}\Vert_{(\rho+1,\infty)}
\]
is finite. Throughout this paper we stand for $\rho_{0}=\frac
{3+\sqrt{17}}{2}>1$ the positive root of the equation $\rho
^{2}-3\rho-2=0$.
From Duhamel's principle, (\ref{SCHd}) is formally equivalent to
the integral equation
\begin{equation}
u(t)=G_{\sigma}(t)u_{0}-i\lambda\int_{0}^{t}G_{\sigma}(t-s)[|u(s)|^{\rho-1}u(s)]ds,
\label{int1}
\end{equation}
where $G_{\sigma}(t)=e^{i(\Delta+\sigma\delta)t}$ is the group determined
by the linear system associated with { (\ref{SCHd})}.
\begin{definition} A mild solution of the initial value problem
(\ref{SCHd}) is a complex-valued function $u\in\mathcal{L}_{\vartheta }^{\infty}$ satisfying (\ref{int1}).
\end{definition}
Our main results of this section read as follows.
\begin{theorem}
\label{GlobalTheo} Let $\sigma\geqq 0$, $\rho_{0}<\rho<\infty$ and $u_{0}\in
\mathcal{E}_{0}.$ There is $\varepsilon>0$ such that {if }$\left\Vert
u_{0}\right\Vert _{\mathcal{E}_{0}}\leq\varepsilon${\ then the IVP
(\ref{SCHd}) has a unique global-in-time mild solution
$u\in\mathcal{L}_{\vartheta }^{\infty}$ satisfying }$\Vert u\Vert_{\mathcal{L}
_{\vartheta }^{\infty}}\leq2\varepsilon.${ } Moreover, the data-solution map
$u_{0}\mapsto u$ from $\mathcal{E}_{0}$ into $\mathcal{L}_{\vartheta }^{\infty}$
is locally Lipschitz.
\end{theorem}
\begin{remark} The proof of Theorem \ref{GlobalTheo} is based in an argument of fixed point, so by using the implicit function theorem is not difficult to show that the data-solution map
$u_{0}\mapsto u$ from $\mathcal{E}_{0}$ into $\mathcal{L}_{\vartheta }^{\infty}$
is smooth.
\end{remark}
\begin{remark}
\label{rem-local}(Local-in-time solutions) Let $1<\rho<\rho_{0}$, $d_{0}=\frac{1}{2}(\frac{\rho
-1}{\rho+1}),$ and $d_{0}<\zeta<\frac{1}{\rho}$. For $0<T<\infty$, consider
the Banach space $\mathcal{L}_{\zeta}^{T}$ of all Bochner measurable functions
$u:(-T,T)\rightarrow$ $L^{(\rho+1,\infty)}$ endowed with the norm
\[
\Vert u\Vert_{\mathcal{L}_{\zeta}^{T}}=\sup_{-T<t<T}|t|^{\zeta}\Vert
u(\cdot,t)\Vert_{(\rho+1,\infty)}.
\]
A local-in-time existence result in $\mathcal{L}_{\zeta}^{T}$ could be proved
for (\ref{SCHd}) by considering $u_{0}\in L^{(\frac{\rho+1}{\rho},\infty)}
(\mathbb{R})$ and small $T>0$ (see \cite{BFV}).
\end{remark}
In the sequel we give an asymptotic stability result for the obtained solutions.
\begin{theorem}
\label{TeoAssin}(Asymptotic Stability) Under the hypotheses of Theorem \ref{GlobalTheo} let $u$ and
$v$ be two solutions of (\ref{int1}) obtained through Theorem \ref{GlobalTheo}
with initial data ${u}_{0}$ and $v_{0},$ respectively. We have that
\begin{equation}
\lim_{\left\vert t\right\vert \rightarrow\infty}\left\vert t\right\vert
^{\vartheta }\left\Vert u(\cdot,t)-v(\cdot,t)\right\Vert _{(\rho+1,\infty)}=0
\label{as1}
\end{equation}
if only if
\begin{equation}
\lim_{\left\vert t\right\vert \rightarrow\infty}\left\vert t\right\vert
^{\vartheta }\left\Vert G_{\sigma}(t)(u_{0}-v_{0})\right\Vert _{(\rho+1,\infty)}=0
\label{as2}
\end{equation}
The condition (\ref{as2}) holds, in particular, for $u_{0}-v_{0}\in
L^{(\frac{\rho+1}{\rho},\infty)}.$
\end{theorem}
| 3,233 | 34,109 |
en
|
train
|
0.38.8
|
\subsection{
Nonlinear Estimate}
In this subsection we give the nonlinear estimate essential in the proof of Theorem \ref{GlobalTheo}. We start by recalling the Beta function
$$
B(\nu,\eta)=\int_{0}^{1}(1-s)^{\nu-1}
s^{\eta-1}ds,
$$
which is finite for all $\nu>0$ and $\eta>0.$ So, for $k_{1}
,k_{2}<1$ and $t>0,$ the change of variable $s\rightarrow st$ yields
\begin{equation}
\int_{0}^{t}(t-s)^{-k_{1}}s^{-k_{2}}ds=t^{1-k_{1}-k_{2}}\int_{0}
^{1}(1-s)^{-k_{1}}s^{-k_{2}}ds=t^{1-k_{1}-k_{2}}B(1-k_{1},1-k_{2})<\infty.
\label{Beta}
\end{equation}
Next we denote by
\begin{equation}
\mathcal{N}(u)=-i\lambda \int_{0}^{t}G_{\sigma}(t-s)[\left\vert u(s)\right\vert ^{\rho-1
}u(s)]ds \label{non1}
\end{equation}
the nonlinear part of the integral equation (\ref{int1}). We have the following estimate in order to apply a point fixed argument.
\begin{lemma}
\label{lem8}Let $\sigma\geqq 0$ and $\rho_{0}<\rho<\infty$. There is a
constant $K>0$ such that
\begin{equation}
\Vert\mathcal{N}(u)-\mathcal{N}(v)\Vert_{\mathcal{L}_{\vartheta }^{\infty}}\leq
K \Vert u-v\Vert_{\mathcal{L}_{\vartheta }^{\infty}}(\Vert u\Vert_{\mathcal{L}
_{\vartheta }^{\infty}}^{\rho-1}+\Vert v\Vert_{\mathcal{L}_{\vartheta }^{\infty}
}^{\rho-1}) \label{est-non1}
\end{equation}
for all $u,v\in\mathcal{L}_{\vartheta }^{\infty}$.
\end{lemma}
\textbf{Proof.} Without loss of generality, we assume $t>0.$ It
follows from (\ref{grupint2}) with $p=\frac{{\rho+1}}{\rho}$, $d=\infty$ and
H\"older inequality (\ref{Holder1}) and $\||f|^r\|_{(p, \infty)}=\|f\|^r_{(rp, \infty)}$ that
\begin{align}
\Vert\mathcal{N}(u)-\mathcal{N}(v)\Vert_{(\rho+1,\infty)} & \leq\int_{0}
^{t}\Vert G_{\sigma}(t-s)(\left\vert u\right\vert ^{\rho-1}u-\left\vert
v\right\vert ^{\rho-1}v)\Vert_{(\rho+1,\infty)}ds\nonumber\\
& \hspace{-3.3cm}\leq C\int_{0}^{t}(t-s)^{-\frac{1}{2}(\frac{2\rho}{\rho
+1}-1)}\Vert(\left\vert u-v\right\vert )(\left\vert u\right\vert ^{\rho
-1}+\left\vert v\right\vert ^{\rho-1})\Vert_{(\frac{{\rho+1}}{\rho}{,\infty)}
}ds\nonumber\\
& \hspace{-3.3cm}\leq C\int_{0}^{t}(t-s)^{-\zeta}\Vert u-v\Vert
_{(\rho+1{,\infty)}}\left( \Vert u\Vert_{(\rho+1{,\infty)}}^{\rho-1}+\Vert
v\Vert_{(\rho+1{,\infty)}}^{\rho-1}\right) ds. \label{aux2}
\end{align}
Next, notice that $\zeta=\frac{1(\rho-1)}{2(\rho+1)}<1$ and $\vartheta \rho<1$ when
$\rho_{0}<\rho<\infty.$ Thus, by using (\ref{Beta}), the r.h.s of
(\ref{aux2}) can be bounded by
\begin{align*}
& \leq C\left( \sup_{0<t<\infty}t^{\vartheta }\Vert u-v\Vert_{(\rho+1{,\infty)}
}\sup_{0<t<\infty}\left( t^{\vartheta (\rho-1)}\Vert u\Vert_{(\rho+1{,\infty)}
}^{\rho-1}+t^{\vartheta (\rho-1)}\Vert v\Vert_{(\rho+1{,\infty)}}^{\rho-1}\right)
\right) \times\int_{0}^{t}(t-s)^{-\zeta}s^{-\vartheta \rho}ds\\
& =CB(1-\zeta, 1-\vartheta \rho) t^{1-\vartheta \rho-\zeta}\left( \Vert u-v\Vert_{\mathcal{L}_{\vartheta
}^{\infty}}(\Vert u\Vert_{\mathcal{L}_{\vartheta }^{\infty}}^{\rho-1}+\Vert
v\Vert_{\mathcal{L}_{\vartheta }^{\infty}}^{\rho-1})\right) ,
\end{align*}
which implies (\ref{est-non1}), because $\zeta+\rho\vartheta =-\vartheta -1.$ \fin
\subsection{Proof of Theorem \ref{GlobalTheo} }
Consider the map $\Phi$ defined on $\mathcal{L}_{\vartheta }^{\infty}$ by
\begin{equation}
\Phi(u)=G_{\sigma}(t)u_{0}+\mathcal{N}(u)\label{map1}
\end{equation}
where $\mathcal{N}(u)$ is given in (\ref{non1}). Let $\mathcal{B}
_{\varepsilon}=\{u\in\mathcal{L}_{\vartheta }^{\infty};\Vert u\Vert_{\mathcal{L}
_{\vartheta }^{\infty}}\leq2\varepsilon\}$ where $\varepsilon>0$ will be chosen
later. Lemma \ref{lem8} implies that
\begin{align}\label{aux}
\Vert\Phi(u)-\Phi(v)\Vert_{\mathcal{L}_{\vartheta }^{\infty}} & =\Vert
\mathcal{N}(u)-\mathcal{N}(v)\Vert_{\mathcal{L}_{\vartheta }^{\infty}}\nonumber\leq K \Vert u-v\Vert_{\mathcal{L}_{\vartheta }^{\infty}}(\Vert u\Vert
_{\mathcal{L}_{\vartheta }^{\infty}}^{\rho-1}+\Vert v\Vert_{\mathcal{L}_{\vartheta
}^{\infty}}^{\rho-1})\\
& \leq2^{\rho}\varepsilon^{\rho-1}K \Vert u-v\Vert_{\mathcal{L}_{\vartheta
}^{\infty}},
\end{align}
for all $u,v\in \mathcal{B}_{\varepsilon}.$ Since
\[
\Vert G_{\sigma}(t)u_{0}\Vert_{\mathcal{L}_{\vartheta }^{\infty}}=\Vert u_{0}
\Vert_{\mathcal{E}_{0}}\leq\varepsilon,
\]
and by using inequality (\ref{est-non1}) with $v=0$ we obtain
\begin{align}
\Vert\Phi(u)\Vert_{\mathcal{L}_{\vartheta }^{\infty}} & \leq\Vert G_{\sigma
}(t)u_{0}\Vert_{\mathcal{L}_{\vartheta }^{\infty}}+\Vert\mathcal{N}(u)\Vert
_{\mathcal{L}_{\vartheta }^{\infty}}\nonumber \leq\Vert G_{\sigma}(t)u_{0}\Vert_{\mathcal{L}_{\vartheta }^{\infty}}+K \Vert
u\Vert_{\mathcal{L}_{\vartheta }^{\infty}}^{\rho}\nonumber\\
& \leq\varepsilon+2^{\rho}\varepsilon^{\rho}K \leq2\varepsilon, \label{aux6}
\end{align}
provided that $2^{\rho}\varepsilon^{\rho-1}K <1$ and $u\in \mathcal{B}_{\varepsilon}.$ It follows that $\Phi:\mathcal{B}_{\varepsilon}\rightarrow \mathcal{B}_{\varepsilon}$ is a contraction, and then it has a fixed point $u\in \mathcal{B}_{\varepsilon},$ $\Phi(u)=u$, which is the unique solution for the integral equation (\ref{int1}) satisfying $\Vert u\Vert
_{\mathcal{\mathcal L}^\infty_{\vartheta }}\leq2\varepsilon.$
In view of (\ref{aux}), if $u,v$ are two integral solutions with respective
data $u_{0},v_{0}$, then
\begin{align*}
\Vert u-v\Vert_{\mathcal{L}_{\vartheta }^{\infty}} & =\Vert G_{\sigma}
(t)(u_{0}-v_{0})\Vert_{\mathcal{L}_{\vartheta }^{\infty}}+\Vert\mathcal{N}
(u)-\mathcal{N}(v)\Vert_{\mathcal{L}_{\vartheta }^{\infty}}\\
& \leq\Vert u_{0}-v_{0}\Vert_{\mathcal{E}_{0}}+2^{\rho}\varepsilon^{\rho
-1}K\Vert u-v\Vert_{\mathcal{L}_{\vartheta }^{\infty}},
\end{align*}
which, due to $2^{\rho}\varepsilon^{\rho-1}K <1,$ yields the Lipschitz
continuity of the data-solution map \fin
\subsection{Proof of Theorem \ref{TeoAssin}}
We will only prove that (\ref{as2}) implies (\ref{as1}). The converse follows
similarly (in fact it is easier) and it is left to the reader. For that matter, we subtract the
integral equations verified by $u$ and $v$ in order to obtain
\begin{align}
t^{\vartheta }\left\Vert u(\cdot,t)-v(\cdot,t)\right\Vert _{(\rho+1,\infty)} &
\leq t^{\vartheta }\left\Vert G_{\sigma}(t)(u_{0}-v_{0})\right\Vert _{(\rho
+1,\infty)}+\nonumber\\
& +t^{\vartheta }\left\Vert \int_{0}^{t}G_{\sigma}(t-s)(\left\vert u\right\vert
^{\rho-1}u-\left\vert v\right\vert ^{\rho-1}v)ds\right\Vert _{(\rho+1,\infty
)}. \label{diff2}
\end{align}
Since $\Vert u\Vert_{\mathcal{L}^\infty_{\vartheta }},\Vert v\Vert_{\mathcal{L}^\infty_{\vartheta }
}\leq2\varepsilon,$ we can estimate the second term in R.H.S. of (\ref{diff2})
as follows.
\begin{align}
& I(t)=t^{\vartheta }\left\Vert\int_{0}^{t}G_{\sigma}(t-s)[\left\vert u\right\vert
^{\rho-1}u-\left\vert v\right\vert ^{\rho-1}v]ds\right\Vert_{(\rho+1,\infty
)}\nonumber\\
& \leq Ct^{\vartheta }\int_{0}^{t}(t-s)^{-\frac{1}{2}(\frac{2\rho}{\rho+1}
-1)}s^{-\vartheta \rho}s^{\vartheta }\Vert u(\cdot,s)-v(\cdot,s)\Vert_{(\rho
+1,\infty)}ds\left( \Vert u\Vert_{\mathcal{L}^\infty_{\vartheta }}^{\rho-1}+\Vert
v\Vert_{\mathcal{L}^\infty_{\vartheta }}^{\rho-1}\right) \nonumber\\
& \leq C2^{\rho}\varepsilon^{\rho-1}t^{\vartheta }\int_{0}^{t}(t-s)^{-\zeta
}s^{-\vartheta \rho}s^{\vartheta }\Vert u(\cdot,s)-v(\cdot,s)\Vert_{(\rho+1,\infty
)}ds, \label{aux9}
\end{align}
where $\zeta=\frac{1}{2}(\frac{2\rho}{\rho+1}-1).$ Now, recalling that
$\zeta+\vartheta \rho-\vartheta -1=-\vartheta ,$ the change of variable $s\longmapsto ts$
in (\ref{aux9}) leads us to
\begin{equation}
I(t)\leq C2^{\rho}\varepsilon^{\rho-1}\int_{0}^{1}(1-s)^{-\zeta}s^{-\vartheta
\rho}(ts)^{\vartheta }\Vert u(\cdot,ts)-v(\cdot,ts)\Vert_{(\rho+1,\infty)}ds.
\label{aux10}
\end{equation}
Set
\begin{equation}
L=\limsup_{t\rightarrow\infty}t^{\vartheta }\Vert u(\cdot,t)-v(\cdot
,t)\Vert_{(\rho+1,\infty)}<\infty\label{defM}
\end{equation}
and recall from proofs of Lemma \ref{lem8} and Theorem \ref{GlobalTheo} that
\[
K =C\int_{0}^{1}(1-s)^{-\zeta}s^{-\vartheta \rho
}ds\quad\text{ and }\quad 2^{\rho}\varepsilon^{\rho-1}K <1.
\]
Then, computing $\limsup_{t\rightarrow\infty}$ in (\ref{diff2}) and using
(\ref{aux10}), we get
\begin{align*}
L & \leq\left( C2^{\rho}\varepsilon^{\rho-1}\int_{0}^{1}(1-s)^{-\zeta
}s^{-\vartheta \rho}ds\right) L\\
& =2^{\rho}\varepsilon^{\rho-1}KL
\end{align*}
and therefore $L=0$, as required. \fin
| 3,295 | 34,109 |
en
|
train
|
0.38.9
|
\section{
Existence of a invariant manifold of periodic orbits}
It is not clear for us whether the approach applied in the proof of Theorem \ref{GlobalTheo} for the case $\mu(x)=\sigma \delta$ in (\ref{SCHd}) with $\sigma\geqq 0$ can be applied for the case $\sigma<0$. Similar situation is happening for the cases $\mu(x)=\alpha(\delta(\cdot-a)+\delta(\cdot+a))$ and $\mu(x)=\beta \delta'$ with $\alpha<0$ and $\beta<0$, respectively.
But, for instance, in the case $\sigma<0$ we can establish a nice qualitative behavior associated to the linear flow generated by equation (\ref{SCHd}). In fact, it follows from Theorem \ref{resol5b} that the linear part of the NLS-$\delta$ equation (\ref{SCHd}) has a two-dimensional manifold of periodic orbits, namely,
$$
E^p=\{\gamma e^{i\theta} \Phi_\sigma(x):\gamma\geqq 0\;\;\text{and}\;\;\theta\in [0, 2\pi]\}.
$$
So, the estimate (\ref{grupint3}) will imply immediately that all solutions $u(t)$ of (\ref{SCHd}) with $\lambda=0$ and with initial conditions $u_0\in L^{(p,d)}(\mathbb R)$ will approach to one of the periodic orbits $\gamma e^{i(\frac{\sigma^2}{4}t+\theta)}\Phi_\sigma\in E^p$. More exactly, we have the following theorem.
\begin{theorem} \label{manifold} Let $\sigma< 0$. For $ d\in [1,\infty]$, $p'\in [1, \infty]$ and $p\in[1,2]$, we have that for $p'$ satisfying $\frac{1}{p}+\frac{1}{p'}=1$, the solution $u(t)$ of the linear equation associated to (\ref{SCHd}) with initial data $u(0)=u_0\in L^{(p,d)}(\mathbb R)$ satisfies
$$
\lim_{t\to \pm \infty} \|u(t)-\gamma_0 e^{i(\frac{\sigma^2}{4}t+\theta)}\Phi_\sigma\|_{(p',d)}=0,
$$
for $\gamma_0=|\langle u_0, \Phi_\sigma\rangle|$ and some $\theta\in [0,2\pi]$.
\end{theorem}
\begin{remark} 1) A similar result to that in Theorem \ref{manifold} can be obtained for the linear equation associated to (\ref{SCHd}) in the case of $\mu(x)=\beta \delta'$ with $\beta<0$ and for the linear equation (\ref{beta3}) in the case of $\mu(x)=\alpha(\delta(\cdot-a)+\delta(\cdot+a))$ with $a\leqq -\frac{1}{\alpha}$ and $\alpha<0$.
2) Note that $\gamma_0<\infty$. In fact, it is not difficult to see that for $
\Psi_\sigma(x)=\sqrt{\frac{-\sigma}{2}}e^{\frac{\sigma}{2}|x|}$ we have for $s\geqq 0$ that $\Psi^*_\sigma(s)=\sqrt{\frac{-\sigma}{2}}e^{\frac{\sigma}{4}s}$. So for all $p, q\in (0,\infty)$ we obtain that
$$
\| \Psi_\sigma\|^q_{L^{(p,q)}}=\int_0^\infty \Big(t^{\frac{1}{p}}\Psi^*_\sigma(t)\Big)^q\frac{dt}{t}= \Big(\frac{-\sigma}{2}\Big)^{\frac{q}{2}}\Big(\frac{-4}{q\sigma}\Big)^{\frac{q}{p}} \Gamma \Big(\frac{q}{p}\Big),
$$
where $\Gamma$ represents the Gamma function. The case $q=\infty$ is immediate. Next, by the Hardy-Littlewood inequality for decreasing rearrangements and the H\"older inequality in the classical $L^p(d\nu)$ spaces, we obtain for $p\in [1,2]$, $p'$ such that $\frac{1}{p}+\frac{1}{p'}=1$ and for $r$ such that $\frac{1}{d}+\frac{1}{r}=1$, with $d\geqq 1$, the estimate
\begin{align*}
\beta_0\leqq \int_{\mathbb R} |u_0(x)|| \Psi_\sigma (x)|dx&\leqq \int_0^\infty u^*_0(t) \Psi^*_\sigma (t)dt=\int_0^\infty t^{\frac{1}{p}}u^*_0(t) t^{\frac{1}{p'}}\Psi^*_\sigma (t)\frac {dt}{t}\\
&\leqq \|u_0\|_{(p,d)}\|\Psi_\sigma\|_{(p',r)}<\infty.
\end{align*}
\end{remark}
| 1,195 | 34,109 |
en
|
train
|
0.38.10
|
\section{
Spaces based on Fourier transform}
\ In this section we consider the nonlinear Schr\"odinger equation
\begin{equation}\label{SCH-F1}
\left\{
\begin{aligned}
i\partial_{t}u+\Delta u+\mu(x)u & =\lambda u^\rho,\ x\in\Omega,\ t\in \mathbb{R},\\
u(x,0) & =u_{0},\text{ }x\in\Omega
\end{aligned}\right.
\end{equation}
where $\mu\in
BC(\mathbb{R}^{n})$ (the space of all bounded continuous functions on $\mathbb{R}^{n})$, $\lambda=\pm1$ \ and $\rho\in\mathbb{N}$. Here we will consider
$\Omega=\mathbb{T}^{n}$ and $\Omega=\mathbb{R}^{n}$, i.e. the periodic and
nonperiodic cases, respectively. The nonlinearity $\lambda\left\vert u\right\vert ^{\rho-1}u$ could be considered in (\ref{SCH-F1}), however we prefer $u^{\rho}$ for the sake of simplicity of
the exposition. For more details, see Remark \ref{rem-Fourier2} below.
We start by defining the spaces for the nonperiodic case. We recall that if $\mathcal{M}(\mathbb{R}^{n})$ denotes the space of complex Radon measures on $\mathbb{R}^{n}$, then it is a vector space and for $\nu\in\mathcal{M}(\mathbb{R}^{n})$, $\|\nu\|_{\mathcal{M}}=|\nu|(\mathbb{R}^{n})$ is a norm on it, where $|\nu|$ is the total variation of $\nu$ (we note that every measure in $\mathcal{M}(\mathbb{R}^{n})$ is automatically a finite Radon measure. Moreover, we can embed $L^1(\mathbb{R}^{n}, dm)$ into $\mathcal{M}(\mathbb{R}^{n})$ by identifying $f\in L^1(\mathbb{R}^{n}, dm)$ with the complex measure $d\nu= fdm$, and $\|\nu\|_{\mathcal{M}}=\int |f|dm$. Next, every $\nu\in \mathcal{M}(\mathbb{R}^{n})$ defines a tempered distribution by $T_\nu(\varphi)=\int_{\mathbb{R}^{n}} \varphi(x)d\nu$, thereby identifying $ \mathcal{M}(\mathbb{R}^{n})$ with a subspace of $\mathcal S'$.
The Fourier transform on $L^1(\mathbb{R}^{n})$ can be extended of a natural form to $\mathcal{M}(\mathbb{R}^{n})$; if $\nu\in \mathcal{M}(\mathbb{R}^{n})$, the Fourier transform of $\nu$ is the function $\widehat{\nu}$ defined by
\begin{equation}\label{transf}
\widehat{\nu}(\xi)=\int e^{-2\pi i \xi\cdot x} d\nu(x),\quad \xi\in \mathbb{R}^{n}.
\end{equation}
Using that $ e^{-2\pi i \xi\cdot x} $ is uniformly continuous in $x$, it is not difficult to check that $\widehat{\nu} \in BC(\mathbb{R}^{n})$ and that $\|\widehat{\nu}\|_{\infty}\leqq \|\nu\|_{\mathcal{M}}$ (see Folland \cite{FO}). Similarly, we define the inverse Fourier transform $\check{\nu}$ of $\nu$ by $\check{\nu}(\xi)=\widehat{\nu}(-\xi)$ for $\xi\in \mathbb{R}^{n}$. Moreover, if $\mathcal F$ represents the Fourier transform on $\mathcal S'$, then for every $\nu\in \mathcal{M}(\mathbb{R}^{n})$ we have $\mathcal F (\nu)=\widehat{\nu}$. Similarly, $\mathcal F^{-1} (\nu)=\check{\nu}$.
Recall that the space $\mathcal{M}(\mathbb{R}^{n})$ can be identified with a subspace of $\mathcal S'$. Hence, if we assume that $ \mu\in \mathcal S'$ and $\mathcal F(\mu)$ is a finite Radon measure, then for $\nu=\mathcal F(\mu)$ we have
$$
\mu=\mathcal F^{-1} \mathcal F(\mu)=\mathcal F^{-1} (\nu)=\check{\nu}\in BC(\mathbb{R}^{n}).
$$
Next, by using the above identification between $\mathcal F$ and\; $\widehat{}$\; \;, we define the Banach space
\begin{equation}
\mathcal{I=}[\mathcal{M}(\mathbb{R}^{n})]^{\vee}=\{f\in\mathcal{S}^{\prime
}(\mathbb{R}^{n}):\widehat{f}\in\mathcal{M}(\mathbb{R}^{n})\}\subset
BC(\mathbb{R}^{n}), \label{esp2}
\end{equation}
with norm
\begin{equation}
\left\Vert f\right\Vert _{\mathcal{I}}=\Vert\widehat{f}\;\Vert
_{\mathcal{M}}.\text{ } \label{norm1}
\end{equation}
We note that in general $\mu\in \mathcal{I}$ may not to belong to $L^{p}(\mathbb{R}^{n})$, nor to $L^{p,\infty}(\mathbb{R}^{n}),$ with $p\neq\infty.$ In particular, $\mu\in \mathcal{I}$ may have infinite $L^{2}$-mass; for instance, if $\mu\equiv1$ then $\mathcal F(\mu)=\delta_{0}\in \mathcal{M}(\mathbb{R}^{n})$.
In the following we will consider $\mu, u_0\in \mathcal{I}$. The Cauchy problem (\ref{SCH-F1}) is formally equivalent to the
functional equation
\begin{equation}
u(t)=S(t)u_{0}+B(u)+L_{\mu}(u), \label{int2}
\end{equation}
where $S(t)=e^{it\Delta}$ is the Schr\"odinger group in $\mathbb{R}^{n}$, and the operators $L_{\mu}(u), B(u)$ are defined via Fourier transform by
\begin{equation}
\widehat{L_{\mu}(u)}=\int_{0}^{t}e^{-i\left\vert \xi\right\vert ^{2}
(t-s)}(\widehat{\mu}\ast\widehat{u})(\xi,s)ds,\label{op1}
\end{equation}
and
\begin{equation}
\widehat{B(u)}=\lambda\int_{0}^{t}e^{-i\left\vert \xi\right\vert ^{2}(t-s)}(\underbrace{\widehat{u}\ast\widehat{u}\ast...\ast\widehat{u}
}_{\rho-times})
(\xi,s)ds, \label{op3}
\end{equation}
for $\mu\in \mathcal{I}$ and $u\in L^{\infty}((-T,T);\mathcal{I})$. We recall that for arbitrary $\mu, \nu\in \mathcal{M}(\mathbb{R}^{n})$ their convolution $\widehat{\mu}\ast\widehat{\nu}\in \mathcal{M}(\mathbb{R}^{n})$ is defined by
$$
\widehat{\mu}\ast\widehat{\nu}(E)=\int\int\chi_E(x+y)d\mu(x)d\nu(y),
$$
for every Borel set $E$.
Let $\mathbb{T}^{n}=\mathbb{R}^{n}/{\mathbb{Z}^{n}}$ stand for the $n$-torus. We say that functions on $\mathbb{T}^{n}$ are functions $f:\mathbb{R}^{n}\to \mathbb{C}$ that satisfy $f(x+m)=f(x)$ for all $x\in \mathbb{R}^{n}$ and $m\in \mathbb{Z}^{n}$, which are called $1$-periodic in every coordinate. Let $\mathcal P=C^\infty_{per}=\{f:\mathbb{R}^{n}\to \mathbb{C}: f \;\text{is}\; C^\infty \;\text{and periodic with period}\; 1\}$. So $\mathcal{D}^{\prime}(\mathbb{T}^{n})$ is the set of all periodic distributions on $\mathcal P$. We say that $\mathcal T:\mathcal P\to \mathbb{C}$ is a {\it periodic distribution} if there exists a sequence $(\Psi_j)_{j\geqq 1}\subset \mathcal P$ such that
$$
\mathcal T(f)=\langle \mathcal T, f\rangle=\lim_{j\to \infty}\int_{[-1/2,1/2]^n} \Psi_j(x)f(x)dx,\qquad \; f\in \mathcal P.
$$
Above we have identify $\mathbb{T}^{n}$ with $[-1/2,1/2]^n$. For a complex-valued function $f\in L^1(\mathbb{T}^{n})$ and $m\in \mathbb{Z}^{n}$, we define
$$
\widehat{f}(m)=\int_{[-1/2,1/2]^n}f(x)e^{-2\pi i x\cdot m}dx.
$$
We call $\widehat{f}(m)$ the $m$-th Fourier coefficient of $f$. The Fourier series of $f$ at $x\in \mathbb{T}^{n}$ is the series
$$
\sum_{m\in \mathbb{Z}^{n}} \widehat{f}(m) e^{2\pi i x\cdot m}.
$$
The Fourier transform of $\mathcal T\in \mathcal{D}^{\prime}(\mathbb{T}^{n})$ is the function $\widehat{\mathcal T}: \mathbb{Z}^{n}\to \mathbb C$ defined by the formula
$$
\widehat{\mathcal T}(m)=\langle \mathcal T, e^{-2\pi i x\cdot m}\rangle,\quad m\in \mathbb{Z}^{n}.
$$
In the periodic case, we are going to study (\ref{SCH-F1}) in the space $\mathcal{I}_{per}$ which is defined by
\begin{equation}
\mathcal{I}_{per}=\{f\in\mathcal{D}^{\prime}(\mathbb{T}^{n}):\widehat{f}\in
l^{1}(\mathbb{Z}^{n})\} \label{peri1}
\end{equation}
endowed with the norm
\begin{equation}
\left\Vert f\right\Vert _{\mathcal{I}_{per}}=\Vert\widehat{f}\;\Vert
_{l^{1}(\mathbb{Z}^{n})}\text{.} \label{peri2}
\end{equation}
Here the IVP (\ref{SCH-F1}) is formally converted to
\begin{equation}
u(t)=S_{per}(t)u_{0}+B_{per}(u)+L_{\mu,per}(u), \label{int-per}
\end{equation}
where, similarly to above, we define the operators in (\ref{int-per}) via
Fourier transform in $\mathcal{D}^{\prime}(\mathbb{T}^{n})$. Precisely, $S_{per}(t)$ is the Schrodinger group in $\mathbb{T}^{n}$
\begin{equation}
S_{per}(t)u_{0}=\sum_{m\in\mathbb{Z}^{n}}\widehat{u}_{0}(m)e^{-4\pi
^{2}i\left\vert m\right\vert ^{2}t}e^{2\pi ix\cdot m}, \label{sch-per}
\end{equation}
\begin{equation}
\widehat{L_{\mu,per}(u)}(m,t)=\int_{0}^{t}e^{-4\pi^{2}i\left\vert m\right\vert
^{2}(t-s)}(\widehat{\mu}\ast\widehat{u})(m,s)ds \label{Op-per1}
\end{equation}
and
\begin{equation}
\widehat{B_{per}(u)}(m,t)=\int_{0}^{t}e^{-4\pi^{2}i\left\vert m\right\vert
^{2}(t-s)}(\underbrace{\widehat{u}\ast\widehat{u}\ast...\ast\widehat{u}
}_{\rho-times})(m,s)ds,
\label{Op-per2}
\end{equation}
for $u_{0},\mu\in \mathcal{I}_{per}$ and $u\in L^{\infty}((-T,T);\mathcal{I}_{per})$, where now the symbol $\ast$ denotes the discrete convolution
\[
\widehat{f}\ast\widehat{g}(m)=\sum_{\xi\in\mathbb{Z}^{n}}\widehat{f}
(m-\xi)\widehat{g}(\xi).
\]
Throughout this section, solutions of (\ref{int2}) or (\ref{int-per}) will be
called mild solutions for the IVP (\ref{SCH-F1}), according the
respective case.
In the above framework, our local-in-time well-posedness result reads as follows.
\begin{theorem}
\label{teoF1}Let $1\leq\rho<\infty.$
\end{theorem}
\begin{itemize}
\item[(1)] (Periodic case) Let $u_{0}\in\mathcal{I}_{per}$ and $\mu$
$\in\mathcal{I}_{per}$. There is $T>0$ such that the IVP (\ref{SCH-F1}) has a unique mild solution $u\in L^{\infty}((-T,T);\mathcal{I}_{per})$ satisfying
\[
\sup_{t\in(-T,T)}\left\Vert u(\cdot,t)\right\Vert _{\mathcal{I}_{per}}
\leq 2\left\Vert u_{0}\right\Vert _{\mathcal{I}_{per}}.
\]
Moreover, the data-map solution $u_{0}\rightarrow u$ is locally Lipschitz continuous
from $\mathcal{I}_{per}$ to {$L^{\infty}((-T,T);$}$\mathcal{I}_{per}\mathcal{)}.$
\item[(2)] (Nonperiodic case) Let $u_{0}\in\mathcal{I}$ and $\mu$
$\in\mathcal{I}$. The same conclusion of item (1) holds true by
replacing $\mathcal{I}_{per}$ by $\mathcal{I}$.
\end{itemize}
\begin{remark}
\label{rem-Fourier}In item (2) of the above theorem, one can show that the
solution $u(x,t)$ verifies $\widehat{u}(\xi,t)\in L^{1}(\mathbb{R}^{n}),$ for
all $t\in(-T,T),$ provided that $\widehat{u}_{0}\in L^{1}(\mathbb{R}^{n})$ and
$\widehat{\mu}\in L^{1}(\mathbb{R}^{n}).$ Then, due to Riemann-Lebesgue lemma,
it follows that
\[
u(x,t)\rightarrow0\text{ as }\left\vert x\right\vert \rightarrow\infty,\text{
for each }t\in(-T,T).
\]
\end{remark}
| 3,557 | 34,109 |
en
|
train
|
0.38.11
|
\subsection{
Nonlinear Estimate}
Before proceeding with the proof of Theorem \ref{teoF1}, let us recall the Young inequality for measures and
discrete convolutions (see Folland \cite{FO} and Iorio\&Iorio \cite{Ior}). For $\mu, \nu \in \mathcal{M}(\mathbb R^n)$ and $f, g\in l^{1}=l^{1}(\mathbb Z^n)$, we have the respective estimates
\begin{align}
\left\Vert \mu\ast \nu\right\Vert _{\mathcal{M}} & \leq\left\Vert \mu\right\Vert
_{\mathcal{M}}\left\Vert \nu\right\Vert _{\mathcal{M}}\label{Young}\\
\left\Vert f\ast g\right\Vert _{l^{1}} & \leq\left\Vert f\right\Vert
_{l^{1}}\left\Vert g\right\Vert _{l^{1}}. \label{Young-per}
\end{align}
\begin{lemma}
\label{lem-est1}Let $1\leq\rho<\infty$ and $0<T<\infty.$
\begin{itemize}
\item[(i)] There exists a positive constant $K>0$ such that
\begin{align}
\sup_{t\in(-T,T)}\left\Vert S_{per}(t)u_{0}\right\Vert _{\mathcal{I}_{per}}
& \leq\left\Vert u_{0}\right\Vert _{\mathcal{I}_{per}}\label{est1}\\
\sup_{t\in(-T,T)}\left\Vert L_{\mu,per}(u-v)\right\Vert _{\mathcal{I}_{per}}
& \leq T\left\Vert \mu\right\Vert _{\mathcal{I}_{per}}\sup_{t\in
(-T,T)}\left\Vert u(\cdot,t)-v(\cdot,t)\right\Vert _{\mathcal{I}_{per}
}\label{est2}\\
\sup_{t\in(-T,T)}\left\Vert B_{per}(u)-B_{per}(v)\right\Vert _{\mathcal{I}
_{per}} & \leq KT\sup_{t\in(-T,T)}\left\Vert u(\cdot,t)-v(\cdot
,t)\right\Vert _{\mathcal{I}_{per}}\label{est3}\\
& \times\left( \sup_{t\in(-T,T)}\left\Vert u(\cdot,t)\right\Vert
_{\mathcal{I}_{per}}^{\rho-1}+\sup_{t\in(-T,T)}\left\Vert v(\cdot
,t)\right\Vert _{\mathcal{I}_{per}}^{\rho-1}\right) ,\nonumber
\end{align}
for all $u_{0},\mu\in\mathcal{I}_{per}$ and $u,v\in${$L^{\infty}((-T,T);$}
$\mathcal{I}_{per}).$
\item[(ii)] The above estimates still hold true with $S(t),L_{\mu}(u),B(u)$
and $\mathcal{I}$ in place of $S_{per}(t),L_{\mu,per}(t),B_{per}(u)$ and
$\mathcal{I}_{per}$, respectively.
\end{itemize}
\end{lemma}
\textbf{Proof.} \ We will only prove the item (i) because (ii) follows similarly
by using (\ref{Young}) instead of (\ref{Young-per}). From definition of
$S_{per}(t),$ we have that
\[
\sup_{t\in(-T,T)}\left\Vert S_{per}(t)u_{0}\right\Vert _{\mathcal{I}_{per}
}=\left\Vert \left( \widehat{u}_{0}(m)e^{-4\pi^{2}i\left\vert m\right\vert
^{2}t}\right) _{m\in\mathbb{Z}^{n}}\right\Vert _{l^{1}(\mathbb{Z}^{n})}
\leq\left\Vert \hat{u}_{0}\right\Vert _{l^{1}(\mathbb{Z}^{n})}.
\]
The operator $L_{\mu,per}$ can be estimated as
\begin{align*}
\left\Vert L_{\mu,per}(u)\right\Vert _{\mathcal{I}_{per}} & =\left\Vert
\widehat{L_{\mu,per}(u)}\right\Vert _{l^{1}(\mathbb{Z}^{n})}\\
& \leq\left\Vert \left( \int_{0}^{t}e^{-4\pi^{2}i\left\vert m\right\vert
^{2}(t-s)}(\widehat{\mu}\ast\widehat{u})(m,s)ds\right) _{m\in\mathbb{Z}^{n}
}\right\Vert _{l_{1}(\mathbb{Z}^{n})}\\
& \leq\sum_{m\in\mathbb{Z}^{n}}\left\vert \int_{0}^{t}e^{-4\pi^{2}i\left\vert
m\right\vert ^{2}(t-s)}(\widehat{\mu}\ast\widehat{u})(m,s)ds\right\vert \\
& \leq\int_{0}^{t}\sum_{m\in\mathbb{Z}^{n}}\left\vert (\widehat{\mu}
\ast\widehat{u})(m,s)\right\vert ds\\
& \leq\int_{0}^{t}\left\Vert \widehat{\mu}\right\Vert _{l^{1}(\mathbb{Z}
^{n})}\left\Vert \widehat{u}(\cdot,s)\right\Vert _{l^{1}(\mathbb{Z}^{n})}ds=\left\Vert \mu\right\Vert _{\mathcal{I}_{per}}\left\Vert u\right\Vert
_{L^{1}(0,T;\mathcal{I}_{per})} \leq T\left\Vert \mu\right\Vert _{\mathcal{I}_{per}}\left\Vert u\right\Vert
_{L^{\infty}(0,T;\mathcal{I}_{per})}.
\end{align*}
By elementary convolution properties and Young inequality (\ref{Young-per}),
it follows that
\begin{align*}
& \left\Vert (\underbrace{\widehat{u}\ast\widehat{u}\ast...\ast\widehat{u}
)}_{\rho-times}-\underbrace{(\widehat{v}\ast\widehat{v}\ast...\ast\widehat{v}
)}_{\rho-times}\right\Vert _{l^{1}(\mathbb{Z}^{n})}\\
& \leq\left\Vert \lbrack(\hat{u}-\hat{v})\ast\widehat{u}\ast...\ast
\widehat{u}+\hat{v}\ast(\widehat{u}-\widehat{v})\ast...\ast\widehat{u}+\hat
{v}\ast\widehat{v}\ast(\widehat{u}-\widehat{v})\ast...\ast\widehat{u}
+...+\hat{v}\ast\widehat{v}\ast...\ast(\widehat{u}-\widehat{v})\right\Vert
_{l^{1}(\mathbb{Z}^{n})}\\
& \leq\left\Vert (\hat{u}-\hat{v})\right\Vert _{l^{1}}\left\Vert \widehat
{u}\right\Vert _{l^{1}}^{\rho-1}+\left\Vert (\hat{u}-\hat{v})\right\Vert
_{l^{1}}\left\Vert \widehat{u}\right\Vert _{l^{1}}^{\rho-2}\left\Vert
\widehat{v}\right\Vert _{l^{1}}+...+\left\Vert (\hat{u}-\hat{v})\right\Vert
_{l^{1}}\left\Vert \widehat{u}\right\Vert _{l^{1}}\left\Vert \widehat
{v}\right\Vert _{l^{1}}^{\rho-2}+\left\Vert (\hat{u}-\hat{v})\right\Vert
_{l^{1}}\left\Vert \widehat{v}\right\Vert _{l^{1}}^{\rho-1}\\
& \leq K\left\Vert (\hat{u}-\hat{v})\right\Vert _{l^{1}}\left( \left\Vert
\widehat{u}\right\Vert _{l^{1}}^{\rho-1}+\left\Vert \widehat{v}\right\Vert
_{l^{1}}^{\rho-1}\right)
\end{align*}
Therefore
\begin{align*}
\left\Vert B_{per}(u)(t)-B_{per}(v)(t)\right\Vert _{\mathcal{I}_{per}} & =\left\Vert
\widehat{B_{per}(u)}-\widehat{B_{per}(v)}\right\Vert _{l^{1}}\\
& \leq\left\Vert \int_{0}^{t}e^{-4\pi^{2}i\left\vert \xi\right\vert
^{2}(t-s)}\left[ (\underbrace{\widehat{u}\ast\widehat{u}\ast...\ast
\widehat{u})}_{\rho-times}-\underbrace{(\widehat{v}\ast\widehat{v}\ast
...\ast\widehat{v})}_{\rho-times}\right] ds\right\Vert _{l^{1}}\\
& \leq\int_{0}^{t}\left\Vert \left[ (\underbrace{\widehat{u}\ast\widehat
{u}\ast...\ast\widehat{u})}_{\rho-times}-\underbrace{(\widehat{v}\ast\widehat
{v}\ast...\ast\widehat{v})}_{\rho-times}\right] \right\Vert _{l^{1}}ds\\
& \leq K\int_{0}^{t}\left\Vert \hat{u}-\hat{v}\right\Vert _{l^{1}}\left(
\left\Vert \widehat{u}\right\Vert _{l^{1}}^{\rho-1}+\left\Vert \widehat
{v}\right\Vert _{l^{1}}^{\rho-1}\right) ds\\
& \leq KT\left\Vert u-v\right\Vert _{L^{\infty}(0,T;\mathcal{I}_{per
}\mathcal{)}}\left( \left\Vert u\right\Vert _{L^{\infty}(0,T;\mathcal{I}
_{per}\mathcal{)}}^{\rho-1}+\left\Vert v\right\Vert _{L^{\infty}
(0,T;\mathcal{I}_{per}\mathcal{)}}^{\rho-1}\right) ,
\end{align*}
as required. \fin
\begin{remark}
\label{rem-Fourier2}
The approach employed here could be used to treat (\ref{SCH-F1}) with
the nonlinearity $\left\vert u\right\vert ^{\rho-1}u$ instead of $u^{\rho}.$
For $\rho$ odd, it would be enough to write $\left\vert u\right\vert
^{\rho-1}u$ (in the above proof) as
\begin{align*}
\lbrack\left( \left\vert u\right\vert ^{2}\right) ^{\frac{\rho-1}{2}
}u]^{\wedge} & =[(u\cdot\overline{u})^{\frac{\rho-1}{2}}u]^{\wedge}\\
& =(\underbrace{\widehat{u}\ast\widehat{u}\ast...\ast\widehat{u})}_{\frac
{\rho-1}{2}-times}\ast(\underbrace{\widehat{\overline{u}}\ast\widehat
{\overline{u}}\ast...\ast\widehat{\overline{u}})}_{\frac{\rho-1}{2}-times}\ast
u
\end{align*}
and to note that $\widehat{\overline{u}}(\xi)=\overline{\widehat{u}}(-\xi)$
and $\left\Vert \overline{u}(\xi)\right\Vert _{\mathcal{I}_{per}}=\left\Vert u(\xi)\right\Vert _{\mathcal{I}_{per}}$.
\end{remark}
| 2,813 | 34,109 |
en
|
train
|
0.38.12
|
\subsection{Proof of Theorem \ref{teoF1}}
\textbf{Proof of (1).} \noindent Consider the ball $\mathcal{B}
_{\varepsilon}=\{u\in L^{\infty}(-T,T;\mathcal{I}_{per});\Vert u\Vert_{L^{\infty
}(-T,T;\mathcal{I}_{per})}\leq2\varepsilon\}$ endowed with the complete metric
$Z(\cdot,\cdot)$ defined by
$$
Z(u,v)=\Vert u-v\Vert_{L^{\infty}(-T,T;\mathcal{I}
_{per})}.
$$
Let $\varepsilon=\Vert u_{0}\Vert_{\mathcal{I}_{per}}$ and $T>0$
such that
\begin{equation}
T(2\left\Vert \mu\right\Vert _{\mathcal{I}_{per}}+2^{\rho}\varepsilon^{\rho
-1}K)<1. \label{cond-small}
\end{equation}
Notice that $\varepsilon$ can be large. We shall show that the map
\begin{equation}
\Phi(u)=S_{per}(t)u_{0}+L_{\mu,per}(u)+B_{per}(u)\nonumber
\end{equation}
\newline is a contraction on $(\mathcal{B}_{\varepsilon},Z).$ Lemma
\ref{lem-est1} with $v=0$ yields
\begin{align}
\Vert\Phi(u)\Vert_{L^{\infty}(-T,T;\mathcal{I}_{per})} & \leq\Vert
S_{per}(t)u_{0}\Vert_{L^{\infty}(-T,T;\mathcal{I}_{per})}+\Vert L_{\mu,per}(u)\Vert_{L^{\infty}(-T,T;\mathcal{I}_{per})}\nonumber\\
&+\left\Vert B_{per}(u)\right\Vert
_{L^{\infty}(-T,T;\mathcal{I}_{per})}\nonumber\\
& \leq\Vert u_{0}\Vert_{\mathcal{I}_{per}}+T\left\Vert \mu\right\Vert
_{\mathcal{I}_{per}}\Vert u\Vert_{L^{\infty}(-T,T;\mathcal{I}_{per})}+TK\Vert
u\Vert_{L^{\infty}(-T,T;\mathcal{I}_{per})}^{\rho}\nonumber\\
& \leq\varepsilon+2\varepsilon T\left\Vert \mu\right\Vert _{\mathcal{I}
_{per}}+2^{\rho}\varepsilon^{\rho}TK\nonumber\\
& =\varepsilon+T(2\left\Vert \mu\right\Vert _{\mathcal{I}_{per}}+2^{\rho
}\varepsilon^{\rho-1}K)\varepsilon<2\varepsilon, \label{aux30}
\end{align}
for all $u\in\mathcal{B}_{\varepsilon}$ and thus $\Phi(\mathcal{B}
_{\varepsilon})\subset\mathcal{B}_{\varepsilon}.$ From Lemma \ref{lem-est1},
we\ also have
\begin{align}
\left\Vert \Phi(u)-\Phi(v)\right\Vert _{L^{\infty}(-T,T;\mathcal{I}_{per})}
& =\Vert L_{\mu,per}(u)-L_{\mu,per}(v)\Vert_{L^{\infty}(-T,T;\mathcal{I}_{per})}\nonumber\\
&+\left\Vert B_{per}(u)-B_{per}(v)\right\Vert _{L^{\infty}(-T,T;\mathcal{I}_{per})}\nonumber\\
& \leq T\left\Vert \mu\right\Vert _{\mathcal{I}_{per}}\Vert u-v\Vert
_{L^{\infty}(-T,T;\mathcal{I}_{per})}\nonumber\\
& +KT\Vert u-v\Vert_{L^{\infty}(-T,T;\mathcal{I}_{per})}\left( \Vert
u\Vert_{L^{\infty}(-T,T;\mathcal{I}_{per})}^{\rho-1}+\Vert v\Vert_{L^{\infty
}(-T,T;\mathcal{I}_{per})}^{\rho-1}\right) \nonumber\\
& \leq T\left( \left\Vert \mu\right\Vert _{\mathcal{I}_{per}}+K2^{\rho
}\varepsilon^{\rho-1}\right) \Vert u-v\Vert_{L^{\infty}(-T,T;\mathcal{I}_{per})}, \label{aux3}
\end{align}
for all $u,v\in\mathcal{B}_{\varepsilon}.$ In view of (\ref{cond-small}),
(\ref{aux30}) and (\ref{aux3}), the map $\Phi$ is a contraction in
$\mathcal{B}_{\varepsilon}$ and then, the Banach fixed point theorem assures
the existence of a unique solution $u\in L^{\infty}(-T,T;\mathcal{I}_{per})$ for
(\ref{int2}) such that $\Vert u\Vert_{L^{\infty}(-T,T;\mathcal{I}_{per})}\leq2\Vert u_0\Vert_{\mathcal{I}_{per}}$.
On the other hand if $u,v$ are two solutions with respective initial data
$u_{0},v_{0}$ then, similarly in deriving (\ref{aux3}), one obtains
\begin{align*}
\Vert u-v\Vert_{L^{\infty}(-T,T;\mathcal{I}_{per})} & \leq\Vert u_{0}-v_{0}\Vert_{L^{\infty
}(-T,T;\mathcal{I}_{per})}+\Vert L_{\mu,per}(u)-L_{\mu,per}(v)\Vert_{L^{\infty}(-T,T;\mathcal{I}_{per})}\nonumber\\
&+\left\Vert B_{per}(u)-B_{per}(v)\right\Vert _{L^{\infty
}(-T,T;\mathcal{I}_{per})}\\
& \leq\Vert u_{0}-v_{0}\Vert_{L^{\infty}(-T,T;\mathcal{I}_{per})}+T\left(
\left\Vert \mu\right\Vert _{\mathcal{I}_{per}}+2^{\rho}\varepsilon^{\rho
-1}K\right) \Vert u-v\Vert_{L^{\infty}(-T,T;\mathcal{I}_{per})},
\end{align*}
which, in view of (\ref{cond-small}), implies the desired local Lipschitz continuity.
\textbf{Proof of (2). }It follows by proceeding entirely parallel to the
proof of item (1) by replacing $\mathcal{I}_{per}$ by $\mathcal{I}.$ \fin
| 1,539 | 34,109 |
en
|
train
|
0.38.13
|
\section{Appendix}
Next we present a different proof of Theorem \ref{expli}. For $\sigma>0$, from Theorem 3.1 in Albeverio {\it et al.} \cite{AGHH} , the fundamental solution $S_\sigma(x, y;t)$ to the Schr\"odinger equation (\ref{SCH1}) is given by
\begin{equation}\label{kern}
S_\sigma(x, y,t)=S(x-y;t)-\frac{\sigma}{2}\int_0^\infty e^{-\frac{\sigma}{2} u} S(u+|x|+|y|;t)du
\end{equation}
where $S(x;t)$ is the free propagator in $\mathbb R$, i.e.
$$
S(x,t)= \frac{e^{-x^2/{4it}}}{(4i\pi t)^{1/2}},\quad t>0
$$
and $e^{it\Delta}f(x)=S(x;t)\ast_x f(x)$. Then we have the representation
\begin{equation}\label{propa}
e^{-itH_\sigma}f(x)=\int_{\mathbb R} S_\sigma(x, y,t)f(y)dy.
\end{equation}
Next, we consider $x>0$ and $f\in L^1(\mathbb R)$ with $supp f\subset (-\infty, 0]$. Then, since
$$
S(u+x-y;t)= (e^{-it\xi^2})^{\vee}(u+x-y)=(e^{-it\xi^2}e^{i\xi(x+u)})^{\vee}(y)
$$
we obtain via Parseval identity that (\ref{propa}) can be re-write for $x>0$ in the form
\begin{equation}\label{propa1}
\begin{aligned}
e^{-itH_\sigma}f(x)&=e^{it\Delta}f(x)\chi^0_+(x) +\int_{\mathbb R}e^{-ity^2}\Big[\int_{\mathbb R} -\frac{\sigma}{2} e^{\frac{\sigma}{2} s}\chi^0_-(s)e^{-iys}ds\Big] \widehat{f}(y)e^{iyx}dy\\
&=e^{it\Delta}f(x)\chi^0_+(x) +\int_{\mathbb R}e^{-ity^2}\widehat{\rho_\sigma\ast f}(y)e^{iyx}dy\\
&=e^{it\Delta}f(x)\chi^0_+(x) +e^{it\Delta}(\rho_\sigma\ast f)(x)\chi^0_+(x)
\end{aligned}
\end{equation}
where $\rho_\sigma(x)=-\frac{\sigma}{2} e^{\frac{\sigma}{2} s}\chi^0_-(s)$. Similarly, for $x<0$ we obtain the representation
\begin{equation}\label{propa2}
e^{itH_\sigma}f(x)=e^{it\Delta}f(x)\chi^0_-(x) +e^{it\Delta}(\rho_\sigma\ast f)(-x)\chi^0_-(x).
\end{equation}
From (\ref{propa1})-(\ref{propa2}) we obtain the formula (\ref{pospro}).
For $\sigma<0$, Theorem 3.1 in \cite{AGHH} establishes that the fundamental solution $S_\sigma(x, y;t)$ to the Schr\"odinger equation (\ref{SCH1}) is given by
$$
S_\sigma(x, y,t)=S(x-y;t) +e^{i\frac{\sigma^2 t}{4}}\Psi_\sigma(x)\Psi_\sigma(y)+\frac{\sigma}{2}\int_0^\infty e^{\frac{\sigma}{2} u} S(u-|x|-|y|;t)du.
$$
where $\Psi_\sigma$ is the (normalized) eigenfunction defined in Theorem \ref{resol5b}. Then, a similar analysis as above produces the formula (\ref{negpro}).
We note that since $|S(x;t)|\leqq C_0 t^{-1/2}$ for every $x\in \mathbb R$ and $t>0$ we obtain from (\ref{kern}) that $|S_\sigma(x,y;t)|\leqq 2C_0 t^{-1/2}$ for all $x, y\in \mathbb R$ and $t>0$. Therefore from (\ref{propa}) we obtain the dispersive estimate
$$
\|e^{-itH_\sigma}f\|_{\infty}\leqq 2C_0 t^{-1/2} \|f\|_1,
$$
which implies the estimate (\ref{stric2}) for $\sigma >0$.
\vskip0.2in
\textbf{ACKNOWLEDGEMENTS:} J. Angulo was partially supported by Grant CNPq/Brazil. L.C.F. Ferreira was supported by FAPESP-SP and CNPq/Brazil.
\end{document}
| 1,205 | 34,109 |
en
|
train
|
0.39.0
|
\begin{document}
\title[On the dimension group of unimodular $\mathcal{S}$-adic subshifts]{On the dimension group\\ of unimodular $\mathcal{S}$-adic subshifts}
\author{V. Berth\'e}
\address{Universit\'e de Paris, IRIF, CNRS, F-75013 Paris, France}
\thanks{This work was supported by the Agence Nationale de la Recherche through the project ``Codys'' (ANR-18-CE40-0007).}
\email{[email protected]}
\author{P. Cecchi Bernales}
\address{Centro de Modelamiento Matemático, Universidad de Chile, Chile}
\thanks{The second author was supported by the PhD grant CONICYT - PFCHA / Doctorado Nacional / 2015-21150544.}
\email{[email protected]}
\author{F. Durand}
\address{LAMFA, UMR 7352 CNRS, Universit\'e de Picardie Jules Verne,
33, rue Saint-Leu, 80039 Amiens, France}
\email{[email protected]}
\author{J. Leroy}
\address{D\'epartement de math\'ematique, Universit\'e de Li\`ege,
12, All\'ee de la d\'ecouverte (B37), 4000 Li\`ege, Belgique}
\email{[email protected]}
\author{D. Perrin}
\address{Laboratoire d'Informatique Gaspard-Monge, Universit\'e Paris-Est, France}
\email{[email protected]}
\author{S. Petite}
\address{LAMFA, UMR 7352 CNRS, Universit\'e de Picardie Jules Verne,
33, rue Saint-Leu, 80039 Amiens, France}
\email{[email protected]}
\date{\today}
\begin{abstract}
Dimension groups are complete invariants of strong orbit equivalence
for minimal Cantor systems.
This paper studies a natural family of minimal Cantor systems having a finitely
generated dimension group, namely the primitive unimodular proper $\mathcal{S}$-adic subshifts. They are generated by iterating sequences
of substitutions. Proper substitutions are such that the images of letters start with a same letter, and similarly end with a same letter.
This family includes various classes of subshifts such as Brun subshifts or dendric subshifts, that in turn include Arnoux-Rauzy subshifts and natural coding of interval exchange transformations.
We compute their dimension group and investigate the relation between
the triviality of the infinitesimal subgroup and rational independence of letter measures. We also introduce the notion of balanced functions and provide a
topological characterization of balancedness for primitive unimodular proper S-adic subshifts.
\end{abstract}
\maketitle
\section{Introduction}
Two dynamical systems are topologically orbit equivalent if there is a homeomorphism between
them preserving the orbits. Originally, the notion of orbit equivalence was studied
in the measurable context (see for instance \cite{Dye,OrnsteinWeiss}), motivated by the classification of von Neumann algebras. In contrast
with the measurable case, Giordano, Putnam and Skau showed that, in the topological setting, uncountably
many classes appear by providing a dimension group as a complete invariant of strong
orbit equivalence \cite{GPS:95}.
Dimension groups are ordered direct limit groups defined by sequences of positive homomorphisms $(\theta _n : \mathbb{Z}^{d_n} \to \mathbb{Z}^{d_{n+1}})_n$,
where $ {\mathbb Z}^d$ is given the standard or simplicial order, and were defined by Elliott \cite{Elliott:76} to study approximately finite dimensional $C^*$-algebras.
In fact, an ordered group is a dimension group if and only if it is a Riesz group \cite{EffrosHandelmanShen:1980}.
They have been widely studied in the late 70's and at the beginning of the 80's \cite{Effros},
in particular when the dimension group is a direct limit given by unimodular matrices \cite{Effros&Shen:1979,Effros&Shen:1980,Effros&Shen:1981,Riedel:1981,Riedel:1981b}.
The present paper studies dynamical and ergodic properties of subshifts having dimension groups
with a group part of the form ${\mathbb Z}^d$.
We focus on the class of primitive unimodular proper $\mathcal{S}$-adic subshifts.
Such subshifts are generated by iterating sequences of substitutions. They have recently attracted much attention in symbolic dynamics
\cite{Berthe&Delecroix:14} and in tiling theory \cite{GM:13,Fusion:14}. Proper substitutions are such that images of letters start with a same letter and also
end with a same letter.
Proper minimal proper $\mathcal{S}$-adic systems have played an important role for the characterization of linearly recurrent subshifts \cite{Durand:2000, Durand:03}.
The term unimodular refers to the unimodularity of the incidence matrices of the associated substitutions.
Sturmian subshifts, subshifts generated by natural codings of interval exchange transformations or Arnoux-Rauzy subshifts are prominent examples
of unimodular proper $\mathcal{S}$-adic subshifts.
They also belong to a recently defined family of subshifts, called dendric subshifts, and considered in \cite{BDFDLPRR:15,BDFDLPRR:15bis,BFFLPR:2015,BDFDLPR:16,Rigidity} (see also Section~\ref{subsec:tree}).
In this series of papers, their elements have been studied under the name of tree words. We have chosen to use the terminology dendric subshift in order to avoid any ambiguity with respect to shifts defined on trees (see, e.g.,~\cite{AubrunBeal}) and also to avoid the puzzling term ``tree word''.
Minimal dendric subshifts are defined with respect to combinatorial properties of their language expressed in terms of extension graphs. For precise definitions, see Section~\ref{subsec:tree}. In particular,
they have linear factor complexity. Focusing on extension properties of factors is a combinatorial viewpoint that allows to highlight the common features shared by dendric subshifts, even if the corresponding symbolic systems have very distinct dynamical, ergodic and spectral properties.
For instance, a coding of a generic interval exchange is topologically weakly mixing for irreducible permutations not of rotation class \cite{NogRud}, whereas an Arnoux-Rauzy subshift is generically not topologically weakly mixing \cite{Cassaigne-Ferenczi-Messaoudi:08,BST:2019}.
Even though one can disprove, in some cases, whether two given minimal dendric subshifts are topologically conjugate by using e.g. asymptotic pairs (see for instance Section \ref{ex:balance}), the question of orbit equivalence is more subtle and is one of the motivations for the present work.
The aim of this paper is to study topological orbit equivalence and strong orbit equivalence for minimal unimodular proper $\mathcal{S}$-adic
subshifts. Let $(X,S)$ be such a subshift over a $d$-letter alphabet ${\mathcal A}$ and
let ${\mathcal M} (X,S)$ stand for its set of shift-invariant probability measures.
One of our main results states that any continuous integer-valued function defined on $X$ is cohomologous to some integer linear combination of characteristic functions of letter cylinders
(Theorem \ref{theo:cohoword}).
This relies on the fact that such subshifts being aperiodic (see Proposition \ref{prop:minimalSSIprimitif}) and recognizable by~\cite{BSTY}, they have a sequence of Kakutani-Rohlin tower partitions
with suitable topological properties.
We then deduce an explicit computation of their dimension group (Theorem \ref{theo:dg}).
Indeed, the dimension group $K^{0}(X,S)$ with ordered unit is isomorphic to
$\left( {\mathbb Z} ^d, \, \{ {\bf x} \in {\mathbb Z}^d \mid \langle {\bf x}, \boldsymbol{ \mu} \rangle > 0 \mbox{ for all }
\mu \in {\mathcal M} (X,S) \}\cup \{{\mathbf 0}\},\, \bf{1}\right )$,
where $\boldsymbol{ \mu}$ denotes the vector of measures of letter cylinders.
In other words,
strong orbit equivalence can be characterized by means of letter measures, i.e., by measures of letter cylinders.
In particular,
two shift-invariant probability measures on $(X,S)$ coinciding on the letter cylinders are proved to be equal (see Corollary \ref{coro:measures}).
This result extends a statement initially proved for interval exchanges in~\cite{FerZam:08}; see also \cite{Putnam:89,Putnam:92,GjerdeJo:02} and \cite{BHL:2019,BHL:2020}.
Moreover, two primitive unimodular proper ${\mathcal S}$-adic subshifts are proved to be strong orbit equivalent if and only if their simplexes of letter measures coincide up to a unimodular matrix (see Corollary \ref{cor:oe}), with the simplex of letter measures being the $d$-simplex generated by the vectors $(\nu[a])_ {a \in \mathcal{A}}$, for $\nu$ in ${\mathcal M}(X,S)$.
We also investigate in Section \ref{sec:saturation} the triviality of the infinitesimal subgroup and relate it to the notion of balance.
We provide
a characterization of the triviality of the infinitesimal subgroup for minimal unimodular proper $\mathcal{S}$-adic
subshifts in terms of rational independence of
measures of letters (see Proposition \ref{theo:saturated}).
Inspired by the classical notion of balance in word combinatorics (see e.g. references in \cite{BerCecchi:2018}),
we also introduce the notion of balanced functions and provide a topological characterization of this balance property for primitive unimodular proper $\mathcal{S}$-adic subshifts (see Corollary \ref{cor:balanced}).
We briefly describe the contents of this paper.
Definitions and basic notions are recalled in Section~\ref{sec:def}, including,
in particular, the notions of dimension group and orbit equivalence in Section~\ref{subsec:dim}, and of image subgroup in Section~\ref{subsec:cyl}. Primitive unimodular $\mathcal{S}$-adic subshifts are introduced in Section~\ref{sec:Sadic}, with dendric subshifts being discussed in more details in Section~\ref{subsec:tree}.
Their dimension groups are studied in Section~\ref{sec:proofs2}. Section~\ref{sec:saturation} is devoted to the study of infinitesimals
and their
connections with the notion of balance. Some examples are handled in Section \ref{sec:examples}.
\paragraph{\bf Acknowledgements}
We would like to thank M. I. Cortez and F. Dolce for stimulating discussions. We also thank warmly the referees of this paper for their careful reading and their
very useful comments, concerning in particular the formulation of Corollary \ref{coro:freq}.
| 3,149 | 37,977 |
en
|
train
|
0.39.1
|
\section{First definitions and background }\label{sec:def}
\subsection{Topological dynamical systems}
By a {\it topological dynamical system}, we mean a pair $(X,T)$
where $X$ is a compact metric space and $T: X \to X$ is a homeomorphism.
It is a {\it Cantor system} when $ X $ is a Cantor space,
that is, $ X $ has a countable basis of its topology which consists of closed and open sets (clopen sets) and does not have isolated points.
This system is {\em aperiodic} if it does not have periodic points, i.e., points $x$ such that $T^n(x) = x$ for some $n >0$.
It is {\em minimal} if it does not contain any non-empty proper closed $T$-invariant subset.
Any minimal Cantor system is aperiodic.
Two topological dynamical systems $(X_1,T_1)$, $(X_2,T_2)$ are {\em conjugate} when there is a {\em conjugacy} between them, i.e., a homeomorphism $\varphi:X_1 \to X_2$ such that $\varphi \circ T_1 = T_2 \circ \varphi$.
A complex number $\lambda$ is a {\em continuous eigenvalue} of
$(X,T)$ if there exists a non-zero continuous function $f \colon X \rightarrow {\mathbb C}$ such that
$f\circ T= \lambda f$.
An {\em additive eigenvalue} is a real number $\alpha$ such that
$\exp(2i \pi \alpha)$ is a continuous eigenvalue.
Let $E(X,T)$ be the (additive) group of additive eigenvalues of $(X,T)$.
We consider its rank over ${\mathbb Q}$, i.e., the maximal number of rationally independent elements of
$E(X,T)$.
Note that $1$ is always an additive eigenvalue and thus ${\mathbb Z} $ is included in $ E(X,T)$.
A probability measure $\mu$ on $X$ is said to be {\em $T$-invariant} if $\mu(T^{-1} A) = \mu(A)$ for every measurable subset $A$ of $ X$.
Let ${\mathcal M} (X,T)$ be the set of all $T$-invariant probability measures on $(X,T)$.
It is a convex set and any extremal point is called an {\em ergodic} $T$-invariant measure. It is well known that any topological dynamical system admits an ergodic invariant measure.
The set of ergodic $T$-invariant probability measures is denoted ${\mathcal M}_e (X,T)$.
Observe that if $(X,T)$ is a minimal Cantor system, then for all clopen $E$ and all $T$-invariant probability measures $\mu$, one has $\mu(E)>0$.
The topological dynamical system $(X,T)$ is {\em uniquely ergodic} if there exists a unique $T$-invariant probability measure on $X$.
It is said to be {\em strictly ergodic} if it is minimal and uniquely ergodic.
The notation $\chi_{E}$ stands for the characteristic function of $E$;
$\mathbb{N}$ stands for the set of non-negative integers ($0 \in \mathbb{N}$).
\subsection{Subshifts} \label{subsec:SD}
Let $\mathcal{A}$ be a finite alphabet of cardinality $d \geq 2$.
Let us denote by $\varepsilon$ the empty word of the free monoid $\mathcal{A}^*$ (endowed with concatenation), and by $\mathcal{A}^{\mathbb{Z}}$ the set of bi-infinite words over $\mathcal{A}$.
For a bi-infinite word $x \in \mathcal{A}^\mathbb{Z}$, and for $i,j \in \mathbb{Z}$ with $i \leq j$, the notation $x_{[i,j)}$ (resp., $x_{[i,j]}$) stands for $x_i \cdots x_{j-1}$ (resp., $x_i \cdots x_{j}$) with the convention $x_{[i,i)} = \varepsilon$.
For a word $w= w_{1} \cdots w_{\ell} \in \mathcal{A}^\ell$,
its {\em length} is denoted $|w|$ and equals $\ell$.
We say that a word $u$ is a {\em factor} of a word $w$ if there exist words $p,s$ such that $w = pus$.
If $p = \varepsilon$ (resp., $s = \varepsilon$) we say that $u$ is a {\em prefix} (resp., {\em suffix}) of $w$.
For a word $u \in \mathcal{A}^{*}$, an index $ 1 \le j \le \ell$ such that $w_{j}\cdots w_{j+|u|-1} =u$ is called an {\em occurence} of $u$ in $w$ and we use the same term for bi-infinite word in $\mathcal{A}^{\mathbb{Z}}$.
The number of occurrences of a word $u \in \mathcal{A}^*$ in a finite word $w$ is denoted as $|w|_u$.
The set $\mathcal{A}^{\mathbb{Z}}$ endowed with the product topology of the discrete topology on each copy of $\mathcal{A}$ is topologically a Cantor set.
The {\em shift map} $S$ defined by $S \left( (x_n)_{n \in \mathbb{Z}} \right) = (x_{n+1})_{n \in \mathbb{Z}}$ is a homeomorphism of $\mathcal{A}^{\mathbb{Z}}$.
A {\em subshift} is a pair $(X,S)$ where $X$ is a closed shift-invariant subset of some $\mathcal{A}^{\mathbb{Z}}$.
It is thus a {\em topological dynamical system}.
Observe that a minimal subshift is aperiodic whenever it is infinite.
The set of factors of a sequence $x \in \mathcal{A}^{\mathbb{Z}}$ is denoted $\mathcal{L}(x)$.
For a subshift $(X,S)$ its {\em language} $\mathcal{L}(X)$ is $\cup_{x\in X} \mathcal{L}(x)$. The {\em factor complexity} $p_X$ of the subshift $(X,S)$ is the function that with $n \in \mathbb{N}$ associates the number $p_X(n)$ of factors of length $n$ in $\mathcal{L}(X)$.
Let $w^-, w^+$ be two words. The cylinder $[w^-.w^+]$ is defined as the set
$\{ x \in X \mid x_{[-|w^-| , |w^+|)} = w^- w^+ \}$.
It is a clopen set.
When $w^-$ is the empty word $\varepsilon$, we set $[\varepsilon .w^+] = [w^+]$.
For $\mu$ a $S$-invariant probability measure, the measure of a factor $w \in \mathcal{L}(X)$ is defined as the measure of the cylinder $[w]$.
The notation $\boldsymbol{ \mu}$ stands for the vector $(\mu([a])_{ a \in {\mathcal A} } \in {\mathbb R}^{\mathcal A}$.
The {\em simplex of letter measures} is defined as the $d$-simplex consisting in all the vectors $\boldsymbol{ \mu}$ with $\mu \in {\mathcal M} (X,S)$, i.e., it consists of all the convex combinations of the vectors $\boldsymbol{ \mu}$ with $\mu \in {\mathcal M}_e (X,S)$.
| 1,848 | 37,977 |
en
|
train
|
0.39.2
|
\subsection{Subshifts} \label{subsec:SD}
Let $\mathcal{A}$ be a finite alphabet of cardinality $d \geq 2$.
Let us denote by $\varepsilon$ the empty word of the free monoid $\mathcal{A}^*$ (endowed with concatenation), and by $\mathcal{A}^{\mathbb{Z}}$ the set of bi-infinite words over $\mathcal{A}$.
For a bi-infinite word $x \in \mathcal{A}^\mathbb{Z}$, and for $i,j \in \mathbb{Z}$ with $i \leq j$, the notation $x_{[i,j)}$ (resp., $x_{[i,j]}$) stands for $x_i \cdots x_{j-1}$ (resp., $x_i \cdots x_{j}$) with the convention $x_{[i,i)} = \varepsilon$.
For a word $w= w_{1} \cdots w_{\ell} \in \mathcal{A}^\ell$,
its {\em length} is denoted $|w|$ and equals $\ell$.
We say that a word $u$ is a {\em factor} of a word $w$ if there exist words $p,s$ such that $w = pus$.
If $p = \varepsilon$ (resp., $s = \varepsilon$) we say that $u$ is a {\em prefix} (resp., {\em suffix}) of $w$.
For a word $u \in \mathcal{A}^{*}$, an index $ 1 \le j \le \ell$ such that $w_{j}\cdots w_{j+|u|-1} =u$ is called an {\em occurence} of $u$ in $w$ and we use the same term for bi-infinite word in $\mathcal{A}^{\mathbb{Z}}$.
The number of occurrences of a word $u \in \mathcal{A}^*$ in a finite word $w$ is denoted as $|w|_u$.
The set $\mathcal{A}^{\mathbb{Z}}$ endowed with the product topology of the discrete topology on each copy of $\mathcal{A}$ is topologically a Cantor set.
The {\em shift map} $S$ defined by $S \left( (x_n)_{n \in \mathbb{Z}} \right) = (x_{n+1})_{n \in \mathbb{Z}}$ is a homeomorphism of $\mathcal{A}^{\mathbb{Z}}$.
A {\em subshift} is a pair $(X,S)$ where $X$ is a closed shift-invariant subset of some $\mathcal{A}^{\mathbb{Z}}$.
It is thus a {\em topological dynamical system}.
Observe that a minimal subshift is aperiodic whenever it is infinite.
The set of factors of a sequence $x \in \mathcal{A}^{\mathbb{Z}}$ is denoted $\mathcal{L}(x)$.
For a subshift $(X,S)$ its {\em language} $\mathcal{L}(X)$ is $\cup_{x\in X} \mathcal{L}(x)$. The {\em factor complexity} $p_X$ of the subshift $(X,S)$ is the function that with $n \in \mathbb{N}$ associates the number $p_X(n)$ of factors of length $n$ in $\mathcal{L}(X)$.
Let $w^-, w^+$ be two words. The cylinder $[w^-.w^+]$ is defined as the set
$\{ x \in X \mid x_{[-|w^-| , |w^+|)} = w^- w^+ \}$.
It is a clopen set.
When $w^-$ is the empty word $\varepsilon$, we set $[\varepsilon .w^+] = [w^+]$.
For $\mu$ a $S$-invariant probability measure, the measure of a factor $w \in \mathcal{L}(X)$ is defined as the measure of the cylinder $[w]$.
The notation $\boldsymbol{ \mu}$ stands for the vector $(\mu([a])_{ a \in {\mathcal A} } \in {\mathbb R}^{\mathcal A}$.
The {\em simplex of letter measures} is defined as the $d$-simplex consisting in all the vectors $\boldsymbol{ \mu}$ with $\mu \in {\mathcal M} (X,S)$, i.e., it consists of all the convex combinations of the vectors $\boldsymbol{ \mu}$ with $\mu \in {\mathcal M}_e (X,S)$.
\subsection{Dimension groups and orbit equivalence}\label{subsec:dim}
Two minimal Cantor systems $(X_1,T_1)$ and $(X_2,T_2)$ are {\em orbit equivalent } if there exists a homeomorphism $\Phi \colon X_1 \rightarrow X_2$
mapping orbits onto orbits, i.e., for all $x \in X_1$, one has
$$\Phi ( \{ T_1^n x \mid n \in {\mathbb Z} \})= \{ T_2^n \Phi (x) \mid n \in {\mathbb Z}\}.$$
This implies that there exist two maps $n_1 \colon X_1 \rightarrow {\mathbb Z}$ and $n_2 \colon X_2 \rightarrow {\mathbb Z}$ (uniquely defined by aperiodicity) such that, for all $x \in X_1$,
$$\Phi \circ T_1 (x)= T_2^{n_1(x)} \circ \Phi(x) \quad \mbox { and } \quad \Phi \circ T_1^{n_2(x)} (x)= T_2 \circ \Phi (x).$$
The minimal Cantor systems $(X_1,T_1)$ and $(X_2,T_2)$ are {\em strongly orbit equivalent} if $n_1$ and $n_2$ both have at most one point of discontinuity.
For more on the subject, see e.g. \cite{GPS:95}.
There is a powerful and convenient way to characterize orbit and strong orbit equivalence in terms of ordered groups and dimension groups due to \cite{GPS:95}.
An {\em ordered group} is a pair $(G,G^+)$, where $G$ is a countable abelian group and $G^+$ is a subset of $G$, called the {\em positive cone}, satisfying
$$
G^+ + G^+ \subset G^+, \quad G^+ \cap (-G^+) = \{ 0 \}, \quad G^+ - G^+ = G.
$$
We write $a\leq b$ if $b-a \in G^+$, and $ a<b$ if $b-a \in G^+$ and $b \neq a$.
An \emph{order ideal}\index{order! ideal} $J$ of an ordered group $(G,G^+)$
is a subgroup $J$ of $G$ such that $J=J^+-J^+$ (with $J^+=J\cap G^+$)
and such that $0\le a\le b\in J$ implies $a\in J$.
An ordered group is \emph{simple}\index{simple ordered group}
\index{ordered group!simple} if it has no nonzero
proper order ideals.
An element $u $ in $G^+$ such that, for all $a $ in $G$, there exists some non-negative integer $n$ with $a \leq nu$ is called an {\em order unit} for $(G,G^+)$.
Two ordered groups with order unit $(G_1,G_1^+,u_1)$ and $(G_2,G_2^+,u_2)$ are {\em isomorphic} when there exists a group isomorphism
$\phi \colon G_1 \rightarrow G_2$ such that $ \phi(G_1^+)=G_2^+$ and $\phi (u_1)=u_2$.
We say that an ordered group is {\em unperforated} if for all $a\in G$, if $na\in G^+$ for some $n\in\mathbb{N}\setminus\{0\}$, then $a\in G^+$.
Observe that this implies in particular that $G$ has no torsion element.
A {\em dimension group} is an unperforated ordered group with order unit $(G,G^+,u)$ satisfying the {\em Riesz interpolation property}: given $a_1, a_2, b_1, b_2\in G$ with $a_i\leq b_j$ ($i,j=1,2$), there exists $c\in G$ with $a_i\leq c\leq b_j$.
Most examples of dimension groups we will deal with in this paper are of the following type:
$(G, G^+,u ) = ( {\mathbb Z} ^d, \, \{ {\bf x}\in {\mathbb Z} ^d \mid \theta_i ( {\bf x}) > 0, \, 1 \leq i \leq e \} \cup\{\mathbf{0}\},u ) $,
where the $\theta_i$'s are independent linear forms such that $\theta_i(u)=1$.
Let $(X,T)$ be a Cantor system.
Let $C(X, {\mathbb R})$ and $C(X, {\mathbb Z})$ respectively stand for the group of continuous functions from $X$ to ${\mathbb R}$ and ${\mathbb Z}$, and let $C(X, {\mathbb N})$ stand for the monoid of continuous functions from $X$ to $\mathbb{N}$, with the group and monoid operation being the addition.
Let
$$
\begin{array}{lrcl}
\beta \colon & C(X, {\mathbb Z}) & \rightarrow & C(X, {\mathbb Z}) \\
& f & \mapsto & f \circ T - f.
\end{array}
$$
A map $f$ is called a {\em coboundary} (resp., a {\em real coboundary}) if there exists a map $g$ in $C(X, {\mathbb Z})$ (resp. in $C(X, {\mathbb R})$) such that $f = g \circ T -g$.
Two maps $f,g \in C(X,\mathbb{Z})$ are said to be {\em cohomologous} whenever $f-g$ is a coboundary.
We consider the quotient group $H(X,T)=C(X, {\mathbb Z})/ \beta C(X,{\mathbb Z})$.
We denote $[f]$ the class of a function $f$ in $H$, and $\pi$ the natural projection $\pi \colon C(X, {\mathbb Z}) \rightarrow H(X,T)$.
We define $H^+ (X, T) = \pi (C(X, {\mathbb N}))$ as the set of classes of functions in $C(X, {\mathbb N})$.
The ordered group with order unit
$$
K^0(X,T) := (H(X,T), H^+ (X,T), [1]),
$$
where $1$ stands for the one constant valued function,
is a dimension group according to \cite{Putnam:89}, called the {\em dynamical dimension group } of $(X,T)$. We will use in this paper
the abbreviated terminology {\em dimension group of $(X,T)$}.
The next result shows that any simple dimension group can be realized as the dimension group of a minimal Cantor system.
\begin{theorem}
\cite[Theorem 5.4 and Corollary 6.3]{HermanPS:92}
\label{theo:simpleminimal}
Let $(G, G^+ , u)$ be a dimension group with order unit.
It is simple if and only if there exists a minimal Cantor system $(X,T)$ such that $(G, G^+ , u)$ is isomorphic to $K^0(X,T)$.
\end{theorem}
We also define the set of {\em infinitesimals} of $K^0 (X,T)$ as
$$
\mbox{Inf} (K^0 (X,T)) = \left \{ [f] \in H(X,T) : \int f d\mu = 0 \mbox { for all } \mu \in {\mathcal M} (X,T)\right\}.
$$
Note that $H(X,T)/\mbox{Inf}(K^0 (X,T))$ with the induced order also determines a dimension group \cite{GPS:95}. We denote it $K^{0} (X,T)/\mbox{\rm Inf} (K^0 (X,T))$.
The dimension groups $K^{0} (X,T)$ and $K^{0} (X,T)/\mbox{Inf} (K^0 (X,T))$ are complete invariants of strong orbit equivalence and orbit equivalence, respectively.
\begin{theorem}\cite{GPS:95}\label{oe}
Let $(X_1,T_1)$ and $(X_2,T_2)$ be two minimal Cantor systems.
The following are equivalent:
\begin{itemize}
\item $(X_1,T_1)$ and $(X_2,T_2)$ are strong orbit equivalent;
\item $K^{0} (X_1,T_1)$ and $K^0 (X_2,T_2)$ are isomorphic.
\end{itemize}
Similarly, the following are equivalent:
\begin{itemize}
\item $(X_1,T_1)$ and $(X_2,T_2)$ are orbit equivalent;
\item $K^{0} (X_1,T_1)/\mbox{\rm Inf} (K^0 (X_1,T_1))$ and $K^{0} (X_2,T_2)/\mbox{\rm Inf} (K^0 (X_2,T_2))$ are isomorphic.
\end{itemize}
\end{theorem}
| 3,320 | 37,977 |
en
|
train
|
0.39.3
|
\subsection{Image subgroup}
\label{subsec:cyl}
A {\em trace} (also called state) of a dimension group $(G,G^+,u)$ is a group homomorphism $p:G\to \mathbb{R}$ such that $p$ is non-negative ($p(G^+)\geq 0$) and $p(u)=1$.
The collection of all traces of $(G,G^+,u)$ is denoted by $\T(G,G^+,u)$.
It is known \cite{Effros} that $\T(G,G^+,u)$ completely determines the order on $G$, if the dimension group is simple.
In fact, one has
\[
G^+=\{a\in G: p(a)>0, \forall p\in \T(G,G^+,u)\}\cup \{0\}.
\]
For more on the subject, see e.g. \cite{Effros}.
Let $(X,T)$ be a minimal Cantor system.
Given $\mu \in {\mathcal M} (X,T)$, we define the {trace} $\tau_{\mu}$ on $K^0 (X,T)$ as $\tau_{\mu} ([f]):= \int f d\mu$.
It is shown in \cite{HermanPS:92} that the correspondence $\mu\mapsto \tau_\mu$ is an affine isomorphism from ${\mathcal M} (X,T)$ onto $\T(K^0(X,T))$.
Thus it sends the extremal points of ${\mathcal M} (X,T)$, i.e., the ergodic measures, to the extremal points of $\T(K^0(X,T))$,
called {\em pure traces}.
The {\em image subgroup} of $K^0 (X,T)$ is defined as the
ordered group with order unit
$$
(I(X,T), I(X,T) \cap {\mathbb R} ^+, 1),
$$
where
$$
I(X,T) = \bigcap_{ \mu \in {\mathcal M} (X,T)} \left\{ \int f d\mu \,: \, f \in C(X,{\mathbb Z})\right\}.
$$
Actually, $E(X,T)$ is a subgroup of $I(X,T)$ (see~\cite[Proposition 11]{CortDP:16} and also \cite[Corollary 3.7]{GHH:18}.
If $(X,T)$ is uniquely ergodic with unique $T$-invariant probability measure $\mu$, then $K^0 (X,T)/\mbox{Inf} (K^0 (X,T))$ is isomorphic to $(I(X,T), I(X,T) \cap {\mathbb R} ^+, 1)$, via the correspondence $$[f]+\mbox{Inf} (K^0 (X,T))\mapsto \int fd\mu.$$
Let us recall the following description of $I(X,T)$.
\begin{proposition}{ \cite[Corollary 2.6]{GHH:18}, \cite[Lemma 12]{CortDP:16}.}
\label{prop:GW}
Let $(X,T)$ be a minimal Cantor system.
Then
$$
I(X,T) = \left\{ \alpha : \exists g \in C(X,\mathbb{Z} ) , \alpha = \int g d \mu \ \forall \mu \in \mathcal{M} (X,T) \right\}.
$$
\end{proposition}
We give an other description of $I(X,T)$ using the following well known lemma.
\begin{lemma}{\cite[Lemma 2.4]{GW}}
\label{lemma:GW}
Let $(X,T)$ be a minimal Cantor system.
Let $f\in C (X, \mathbb{Z} )$ such that $\int_X f d \mu $ belongs to $]0,1[$ for every $\mu \in \mathcal{M} (X , T)$.
Then, there exists a clopen set $U$ such that $\int_X f d \mu = \mu (U)$ for every $\mu \in \mathcal{M} (X , T)$.
\end{lemma}
For a family of real
numbers $N = \{\alpha_{i} : i \in J\}$, we let $\langle N \rangle$ denote the abelian additive group generated by these real numbers.
\begin{proposition}\label{prop:cob}
Let $(X,T)$ be a minimal Cantor system.
Then,
\begin{equation}\label{eq:I}
I(X,T)
= \left\langle \{ \alpha : \exists \ \hbox{\rm clopen set } U \subset X , \ \alpha = \mu (U) \ \forall \mu \in \mathcal{M}(X,T)\}\right\rangle.
\end{equation}
\end{proposition}
\begin{proof}
There is just one inclusion to prove.
Let $\beta$ be in $I(X , T )$.
If $\beta$ is an integer then using that $\beta = \beta\mu (X)$ for all $\mu \in \mathcal{M} (X,T)$, it implies that $\beta$ belongs to the right member of the equality in \eqref{eq:I}.
Otherwise, let $n\in \mathbb{Z}$ be such that $\beta-n$ belongs to $]0,1[$.
From Proposition \ref{prop:GW} and Lemma \ref{lemma:GW} there exists a clopen set $U$ such that $\mu (U) = \beta-n$ for all $\mu \in \mathcal{M} (X,T)$.
It follows $\mu (U)$ and $n$ belong to
$\left\langle \{ \alpha : \exists \ \hbox{\rm clopen set } U \subset X , \ \alpha = \mu (U) \ \forall \mu \in \mathcal{M}(X,T)\}\right\rangle $. So it is also the case for $\beta$.
\end{proof}
One can obtain a more explicit description of the set $I (X,S)$ for minimal
subshifts.
\begin{proposition}\label{prop:cob}
Let $(X,S)$ be a minimal subshift.
Then,
$$\begin{array}{ll}
I(X,S)&=\bigcap_{\mu\in \mathcal{M}(X,S)} \left\langle \{\mu([w]): w\in\mathcal{L}(X)\}\right\rangle .
\end{array}$$
In particular, if $(X,S)$ is uniquely ergodic with $\mu$ its unique $S$-invariant probability measure, then
$I(X,S)=\left\langle \{\mu([w]): w\in\mathcal{L}(X)\}\right\rangle.$
\end{proposition}
\begin{proof}
The proof of the first equality is a direct consequence of the fact that every function belonging to $C(X,\mathbb{Z})$ is cohomologous to some cylinder function in $C(X,\mathbb{Z})$, i.e.,
to some function $h$ in $ C(X,\mathbb{Z})$ for which there exists $n>0$ such that for all $x\in X$, $h(x)$ depends only on $x_{[0,n)}$.
Indeed, let $f\in C(X,\mathbb{Z})$. Since $f$ is integer-valued, it is locally constant, and by compactness of $X$, there exists $k\in\mathbb{N}$ such that for all $x\in X$, $f(x)$ depends only on $x_{[-k,k]}$. Therefore, $g(x)=f\circ S^k(x)$ belongs to $C(X,\mathbb{Z})$ and depends only on $x_{[0,2k]}$ for all $x\in X$.
Finally, $f-g=f-f\circ S^k(x)= f- f\circ S+ f\circ S - f \circ S^2 + \cdots + f\circ S^{k-1}(x)+ f\circ S^{k}(x) $ is a coboundary.
Hence, $\int fd\mu=\int gd\mu$. Since $g$ is a cylinder function, $g$ can be written as a finite sum of the form
$g=\sum\ell_u\chi_{[u]},$ $u\in \mathcal{L}(X)$ and $\ell_u\in\mathbb{Z}$. Thus,
$\int f d\mu=\sum_{}\ell_u\mu([u])\in \left\langle \{\mu([w]): w\in\mathcal{L}(X)\}\right\rangle.$
\end{proof}
| 2,138 | 37,977 |
en
|
train
|
0.39.4
|
\section{Primitive unimodular proper $\mathcal{S}$-adic subshifts}\label{sec:Sadic}
In this section we first recall the notion of primitive unimodular proper $\mathcal{S}$-adic subshift in Section~\ref{subsec:sadic}.
We then illustrate it with the class of minimal dendric subshifts in Section \ref{subsec:tree}.
| 101 | 37,977 |
en
|
train
|
0.39.5
|
\subsection{$\mathcal{S}$-adic subshifts} \label{subsec:sadic}
Let $\mathcal{A},\, \mathcal{B}$ be finite alphabets and let $\tau:\, \mathcal{A}^*\to \mathcal{B}^*$ be a \emph{non-erasing} morphism (also called a \emph{substitution} if $\mathcal{A} = \mathcal{B}$). Let us note that a morphism is uniquely determined by its values on the alphabet ${\mathcal A}$
and this will be the way we will define them (see e.g. Example \ref{ex:fibo}). By non-erasing, we mean that the image of any letter is a non-empty word. We stress the fact that all morphisms are assumed to be non-erasing in the following.
Using concatenation, we extend $\sigma$ to~$\mathcal{A}^\mathbb{N}$ and~$\mathcal{A}^\mathbb{Z}$.
With a morphism $\tau :\mathcal{A}^* \to \mathcal{B}^*$, where $\mathcal{A}$ and $\mathcal{B}$ are finite alphabets, we classically associate an {\em incidence matrix} $M_\tau$ indexed by $\mathcal{B} \times \mathcal{A}$ such that for every $(b,a)\in \mathcal{B} \times \mathcal{A}$, its entry at position $(b,a)$ is the number of occurrences of $b$ in $\tau(a)$.
Alphabets are always assumed to have cardinality at least 2.
The morphism $\tau$ is said to be {\em left proper} (resp. {\em right proper}) when there exist a letter $b \in \mathcal{B}$ such that for all $a \in \mathcal{A}$, $\tau(a)$ starts with $b$ (resp., ends with $b$).
It is said to be {\em proper} if it is both left and right proper.
We recall the definition of an $\mathcal{S}$-adic subshift as stated in \cite{BSTY}, see also \cite{Berthe&Delecroix:14} for more on $\mathcal{S}$-adic subshifts.
Let $\boldsymbol{\tau} = (\tau_n : \mathcal{A}_{n+1}^* \to \mathcal{A}_n^*)_{n \geq 1}$ be a sequence of morphisms such that
$
\max_{a \in \mathcal{A}_n} |\tau_1 \circ \cdots \circ \tau_{n-1}(a)|
$
goes to infinity when $n$ increases. By non-erasing, we mean that the image of any letter is a non-empty word.
For $1\leq n<N$, we define $\tau_{[n,N)} = \tau_n \circ \tau_{n+1} \circ \dots \circ \tau_{N-1}$ and $\tau_{[n,N]} = \tau_n \circ \tau_{n+1} \circ \dots \circ \tau_{N}$.
For $n\geq 1$, the \emph{language $\mathcal{L}^{(n)}({\boldsymbol{\tau}})$ of level $n$ associated with $\boldsymbol{\tau}$} is defined~by
\[
\mathcal{L}^{(n)}({\boldsymbol{\tau}}) = \big\{w \in \mathcal{A}_n^* \mid \mbox{$w$ occurs in $\tau_{[n,N)}(a)$ for some $a \in\mathcal{A}_N$ and $N>n$}\big\}.
\]
As $\max_{a \in \mathcal{A}_n} |\tau_{[1,n)}(a)|$ goes to infinity when $n$ increases, $\mathcal{L}^{(n)}({\boldsymbol{\tau}})$ defines a non-empty subshift $X_{\boldsymbol{\tau}}^{(n)}$ that we call the {\em subshift generated by $\mathcal{L}^{(n)}({\boldsymbol{\tau}})$}.
More precisely, $X_{\boldsymbol{\tau}}^{(n)}$ is the set of points $x \in \mathcal{A}_n^\mathbb{Z}$ such that $\mathcal{L} (x) \subseteq \mathcal{L}^{(n)}({\boldsymbol{\tau}})$. Note that it may happen that $\mathcal{L}(X_{\boldsymbol{\tau}}^{(n)})$ is strictly contained in $\mathcal{L}^{(n)}({\boldsymbol{\tau}})$.
We set $\mathcal{L}({\boldsymbol{\tau}}) = \mathcal{L}^{(1)}({\boldsymbol{\tau}})$,$X_{\boldsymbol{\tau}} = X_{\boldsymbol{\tau}}^{(1)}$ and call $(X_{\boldsymbol{\tau}},S)$ the \emph{$\mathcal{S}$-adic subshift} generated by the \emph{directive sequence}~$\boldsymbol{\tau}$.
We say that $\boldsymbol{\tau}$ is {\em primitive} if, for any $n\geq 1$, there exists $N>n$ such that
$M_{\tau_{[n,N)}} >0$, {\em i.e.}, for all $a \in \mathcal{A}_N$, $\tau_{[n,N)}(a)$ contains occurrences of all letters of $\mathcal{A}_{n}$.
Of course, $M_{\tau_{[n,N)}}$ is equal to $M_{\tau_n} M_{ \tau_{n+1}}\cdots M_{\tau_{N-1}}$.
Observe that if $\boldsymbol{\tau}$ is primitive, then $\min_{a \in \mathcal{A}_n} |\tau_{[1,n)}(a)|$ goes to infinity when $n$ increases.
In the primitive case $\mathcal{L}(X_{\boldsymbol{\tau}}^{(n)})= \mathcal{L}^{(n)}({\boldsymbol{\tau}})$, and $X_{\boldsymbol{\tau}}^{(n)}$ is a minimal subshift (see for instance ~\cite[Lemma 7]{Durand:2000}).
We say that $\boldsymbol{\tau}$ is ({\em left, right}) {\em proper} whenever each morphism $\tau_n$ is (left, right) proper.
We also say that $\boldsymbol{\tau}$ is {\em unimodular} whenever, for all $n \geq 1$, $\mathcal{A}_{n+1} = \mathcal{A}_n$ and the matrix $M_{\tau_n}$ has determinant of absolute value 1.
By abuse of language, we say that a subshift is a (unimodular, left or right proper, primitive) $\mathcal{S}$-adic subshift if there exists a (unimodular, left or right proper, primitive) sequence of morphisms $\boldsymbol{\tau}$ such that $X = X_{\boldsymbol{\tau}}$.
Let us give another way to define $X_{\boldsymbol{\tau}}$ when $\boldsymbol{\tau}$ is primitive and proper.
For an endomorphism $\tau$ of $\mathcal{A}^*$, let $\Omega (\tau ) = \overline{\bigcup_{k\in \mathbb{Z}} S^k \tau ( \mathcal{A}^\mathbb{Z} )}$.
\begin{lemma}
\label{lemma:omega}
Let $\boldsymbol{\tau} = (\tau_n : \mathcal{A}_{n+1}^* \to \mathcal{A}_n^*)_{n \geq 1}$ be a sequence of morphisms such that $\min_{a \in \mathcal{A}_n} |\tau_{[1,n)}(a)|$ goes to infinity when $n$ increases.
Then,
$$
X_{\boldsymbol{\tau}} \subset \bigcap_{n\in \mathbb{N}} \Omega (\tau_{[1,n]} ).
$$
Furthermore, when $\boldsymbol{\tau}$ is primitive and proper, then the equality $
X_{\boldsymbol{\tau}} = \bigcap_{n\in \mathbb{N}} \Omega (\tau_{[1,n]} )
$ holds.
\end{lemma}
\begin{proof}
The proof is left to the reader.
\end{proof}
With a left proper morphism $\sigma:\mathcal{A}^* \to \mathcal{B}^*$ such that $b \in \mathcal{B}$ is the first letter of all images $\sigma(a)$, $a \in \mathcal{A}$, we associate the right proper morphism $\overline{\sigma}:\mathcal{A}^* \to \mathcal{B}^*$ defined by
$b\overline{\sigma}(a) = \sigma(a)b$ for all $a \in \mathcal{A}$.
For all $x \in A^\mathbb{Z}$, we thus have $\bar \sigma(x) = S \sigma(x)$.
The next result is a weaker version of~\cite[Corollary 2.3]{Durand&Leroy:2012}.
\begin{lemma}
\label{lemma:proper}
Let $(X,S)$ be an $\mathcal{S}$-adic subshift generated by the
primitive and left proper directive sequence $\boldsymbol{\tau} = (\tau_n:\mathcal{A}_{n+1}^* \to \mathcal{A}_n^*)_{n \geq 1}$.
Then $(X,S)$ is also generated by
the primitive and proper directive sequence $\tilde{\boldsymbol{\tau}} = (\tilde{\tau}_n)_{n \geq 1}$, where for all $n$, $\tilde{\tau}_n = \tau_{2n-1} \overline{\tau}_{2n}$.
In particular, if $\boldsymbol{\tau}$ is unimodular, then so is $\tilde{\boldsymbol{\tau}}$.
\end{lemma}
\begin{proof}
Each morphism $\tilde{\tau}_n$ is trivially proper.
It is also clear that the unimodularity and the primitiveness of $\boldsymbol{\tau}$ are preserved in this process.
Using the relation $\bar \sigma(x) = S \sigma(x)$ and Lemma~\ref{lemma:omega}, we then get
\[
X_{\boldsymbol{\tau}}
\subset \bigcap_{n \in \mathbb{N}} \Omega (\tau_{[1,n]})
= \bigcap_{n \in \mathbb{N}} \Omega (\tilde{\tau}_{[1,n]})
= X_{\tilde{\boldsymbol{\tau}}}.
\]
Since both $\boldsymbol{\tau}$ and $\tilde{\boldsymbol{\tau}}$ are primitive, the subshifts $X_{\boldsymbol{\tau}}$ and $X_{\tilde{\boldsymbol{\tau}}}$ are minimal, hence they are equal.
\end{proof}
\begin{lemma}
\label{lemma:aperiodicity}
All primitive unimodular proper $\mathcal{S}$-adic subshifts are aperiodic.
\end{lemma}
\begin{proof}
Let $\boldsymbol{\tau}$ be a primitive unimodular proper directive sequence on the alphabet $\mathcal{A}$ of cardinality $d \geq 2$.
Suppose that it has a periodic point $x$ of period $p$, where $p$ is the smallest period of $x$ ($p>0$).
By primitiveness, all letters of $\mathcal{A}$ occur in $x$, so $p \geq d$.
We have $x = \cdots uu.uu\cdots $ for some word $u$ with $|u| = p$.
There exists some $n$ such that, for all $a $, one has $\tau_{[1,n]} (a) = s(a) u^{q(a)}p(a)$, where $s(a), p(a)$ are a strict suffix and a strict prefix of $u$ and $q(a) >1$.
Let $b\in \mathcal{A}$ and set $\tau_{n+1} (b) = b_0 b_1 \cdots b_k$.
As the directive sequence $\boldsymbol{\tau}$ is proper, $b_0 b_1 \cdots b_k b_0$ is also a word in $\mathcal{L}^{(n+1)} (\boldsymbol{\tau})$.
By a classical argument due to Fine and Wilf \cite{Fine:65}, one has $p(b_0)s(b_1) = p(b_i)s(b_{i+1})= p(b_k)s(b_0) = u$ for $1\leq i\leq k-1$.
Hence
$$
|\tau_1 \cdots \tau_{n+1} (b) |
\equiv
|s(c)p(c)s(b_1)p(b_1) \cdots s(b_k)p(b_k)|
\equiv
0 \mbox{ modulo } |u|,
$$
which contradicts the unimodularity of $\boldsymbol{\tau}$.
\end{proof}
| 2,966 | 37,977 |
en
|
train
|
0.39.6
|
With a left proper morphism $\sigma:\mathcal{A}^* \to \mathcal{B}^*$ such that $b \in \mathcal{B}$ is the first letter of all images $\sigma(a)$, $a \in \mathcal{A}$, we associate the right proper morphism $\overline{\sigma}:\mathcal{A}^* \to \mathcal{B}^*$ defined by
$b\overline{\sigma}(a) = \sigma(a)b$ for all $a \in \mathcal{A}$.
For all $x \in A^\mathbb{Z}$, we thus have $\bar \sigma(x) = S \sigma(x)$.
The next result is a weaker version of~\cite[Corollary 2.3]{Durand&Leroy:2012}.
\begin{lemma}
\label{lemma:proper}
Let $(X,S)$ be an $\mathcal{S}$-adic subshift generated by the
primitive and left proper directive sequence $\boldsymbol{\tau} = (\tau_n:\mathcal{A}_{n+1}^* \to \mathcal{A}_n^*)_{n \geq 1}$.
Then $(X,S)$ is also generated by
the primitive and proper directive sequence $\tilde{\boldsymbol{\tau}} = (\tilde{\tau}_n)_{n \geq 1}$, where for all $n$, $\tilde{\tau}_n = \tau_{2n-1} \overline{\tau}_{2n}$.
In particular, if $\boldsymbol{\tau}$ is unimodular, then so is $\tilde{\boldsymbol{\tau}}$.
\end{lemma}
\begin{proof}
Each morphism $\tilde{\tau}_n$ is trivially proper.
It is also clear that the unimodularity and the primitiveness of $\boldsymbol{\tau}$ are preserved in this process.
Using the relation $\bar \sigma(x) = S \sigma(x)$ and Lemma~\ref{lemma:omega}, we then get
\[
X_{\boldsymbol{\tau}}
\subset \bigcap_{n \in \mathbb{N}} \Omega (\tau_{[1,n]})
= \bigcap_{n \in \mathbb{N}} \Omega (\tilde{\tau}_{[1,n]})
= X_{\tilde{\boldsymbol{\tau}}}.
\]
Since both $\boldsymbol{\tau}$ and $\tilde{\boldsymbol{\tau}}$ are primitive, the subshifts $X_{\boldsymbol{\tau}}$ and $X_{\tilde{\boldsymbol{\tau}}}$ are minimal, hence they are equal.
\end{proof}
\begin{lemma}
\label{lemma:aperiodicity}
All primitive unimodular proper $\mathcal{S}$-adic subshifts are aperiodic.
\end{lemma}
\begin{proof}
Let $\boldsymbol{\tau}$ be a primitive unimodular proper directive sequence on the alphabet $\mathcal{A}$ of cardinality $d \geq 2$.
Suppose that it has a periodic point $x$ of period $p$, where $p$ is the smallest period of $x$ ($p>0$).
By primitiveness, all letters of $\mathcal{A}$ occur in $x$, so $p \geq d$.
We have $x = \cdots uu.uu\cdots $ for some word $u$ with $|u| = p$.
There exists some $n$ such that, for all $a $, one has $\tau_{[1,n]} (a) = s(a) u^{q(a)}p(a)$, where $s(a), p(a)$ are a strict suffix and a strict prefix of $u$ and $q(a) >1$.
Let $b\in \mathcal{A}$ and set $\tau_{n+1} (b) = b_0 b_1 \cdots b_k$.
As the directive sequence $\boldsymbol{\tau}$ is proper, $b_0 b_1 \cdots b_k b_0$ is also a word in $\mathcal{L}^{(n+1)} (\boldsymbol{\tau})$.
By a classical argument due to Fine and Wilf \cite{Fine:65}, one has $p(b_0)s(b_1) = p(b_i)s(b_{i+1})= p(b_k)s(b_0) = u$ for $1\leq i\leq k-1$.
Hence
$$
|\tau_1 \cdots \tau_{n+1} (b) |
\equiv
|s(c)p(c)s(b_1)p(b_1) \cdots s(b_k)p(b_k)|
\equiv
0 \mbox{ modulo } |u|,
$$
which contradicts the unimodularity of $\boldsymbol{\tau}$.
\end{proof}
The next two results will be important for the computation of the dimension group of primitive unimodular proper $\mathcal{S}$-adic subshifs.
The first one is a weaker version of~\cite[Theorem 3.1]{BSTY}.
\begin{theorem}[\cite{BSTY}]
\label{theo:BSTY}
Let $\tau :\mathcal{A}^* \to \mathcal{B}^*$ be such that its incidence matrix $M_\tau$ is unimodular.
Then, for any aperiodic $y \in \mathcal{B}^\mathbb{Z}$, there exists at most one $(k,x) \in \mathbb{N} \times A^\mathbb{Z}$ such that $y = S^k \tau(x)$, with $0 \leq k < |\tau(x_0)|$.
\end{theorem}
\begin{proposition}
\label{prop:minimalSSIprimitif}
Let $\boldsymbol{\tau} = (\tau_n : \mathcal{A}^* \to \mathcal{A}^*)_{n \geq 1}$ be a unimodular proper sequence of morphisms such that $\max_{a \in \mathcal{A}} |\tau_{[1,n)}(a)|$ goes to infinity when $n$ increases.
Then $(X_{\boldsymbol{\tau}},S)$ is aperiodic and minimal if and only if $\boldsymbol{\tau}$ is primitive.
\end{proposition}
\begin{proof}
Recall that any $\mathcal{S}$-adic subshift with a primitive directive sequence is minimal (see, e.g.~\cite[Lemma 7]{Durand:2000}) and that aperiodicity is proved in Lemma \ref{lemma:aperiodicity}.
We only have to show that the condition is necessary.
We assume that $(X_{\boldsymbol{\tau}},S)$ is aperiodic and minimal.
For all $n \geq 1$, $(X_{\boldsymbol{\tau}}^{(n)},S)$ is trivially aperiodic.
Let us show that it is minimal.
Assume by contradiction that for some $n \geq 1$, $(X_{\boldsymbol{\tau}}^{(n)},S)$ is minimal, but not $(X_{\boldsymbol{\tau}}^{(n+1)},S)$.
There exist $u \in \mathcal{L}(X_{\boldsymbol{\tau}}^{(n+1)})$ and $x \in X_{\boldsymbol{\tau}}^{(n+1)}$ such that $u$ does not occur in $x$.
By Theorem~\ref{theo:BSTY}, $\{\tau_n([v]) \mid v \in \mathcal{L}(X_{\boldsymbol{\tau}}^{(n+1)}) \cap \mathcal{A}^{|u|}\}$ is a finite clopen partition of
$\tau_n(X_{\boldsymbol{\tau}}^{(n+1)})$.
Thus, considering $y = \tau_n(x)$, by minimality of $(X_{\boldsymbol{\tau}}^{(n)},S)$, there exists $k \geq 0$ such that $S^k y$ is in $\tau_n([u])$.
Take $z \in [u]$ such that $S^ky = \tau_n(z)$.
Since $y$ is aperiodic and since we also have $S^k y = S^{k'} \tau_n(S^\ell x)$ for some $\ell \in \mathbb{N}$ and $0 \leq k' < |\tau_n(x_\ell)|$,
we obtain that $\tau_n(z) = S^{k'} \tau_n(S^\ell x)$ with $z \in [u]$, $S^\ell x \notin [u]$ and $0 \leq k' < |\tau_n(x_\ell)|$; this contradicts Theorem~\ref{theo:BSTY}.
We now show that $\lim_{n \to +\infty} \min_{a \in \mathcal{A}} |\tau_{[1,n)}(a)| = +\infty$.
We again proceed by contradiction, assuming that $(\min_{a \in \mathcal{A}} |\tau_{[1,n)}(a)|)_{n \geq 1}$ is bounded.
Then there exists $N > 0$ and a sequence $(a_n)_{n \geq N}$ of letters in $\mathcal{A}$ such that for all $n \geq N$, $\tau_n(a_{n+1}) = a_n$.
We claim that there are arbitrary long words of the form $a_N^k$ in $\mathcal{L}(X_{\boldsymbol{\tau}}^{(N)})$ which contradicts the fact that $(X_{\boldsymbol{\tau}}^{(N)},S)$ is minimal and aperiodic.
Since $\boldsymbol{\tau}$ is proper, for all $n \geq N$ and all $b \in \mathcal{A}$, $\tau_n(b)$ starts and ends with $a_n$.
As $\max_{a \in \mathcal{A}} |\tau_{[1,n)}(a)|$ goes to infinity, there exists a sequence $(b_n)_{n \geq N}$ of letters in $\mathcal{A}$ such that $|\tau_{[N,n)}(b_n)|$ goes to infinity and for all $n \geq N$, $b_n$ occurs in $\tau_n(b_{n+1})$.
This implies that there exists $M \geq N$ such that for all $n \geq M$, $b_n \neq a_n$ and, consequently, that $\tau_n(b_{n+1}) = a_n u_n$ for some word $u_n$ containing $b_n$.
It is then easily seen that, for all $k \geq 1$, $a_M^k$ is a prefix of $\tau_{[M,M+k)}(b_{M+k})$, which proves the claim.
We finally show that $\boldsymbol{\tau}$ is primitive.
If not, there exist $N \geq 1$ and a sequence $(a_n)_{n \geq N}$ of letters in $\mathcal{A}$ such that for all $n > N$, $a_N$ does not occur in $\tau_{[N,n)}(a_n)$.
As $(|\tau_{[N,n)}(a_n)|)_n$ goes to infinity, this shows that there are arbitrarily long words in $\mathcal{L}(X_{\boldsymbol{\tau}}^{(N)})$ in which $a_N$ does not occur.
Since $\boldsymbol{\tau}$ is unimodular, there is also a sequence $(a'_n)_{n \geq N}$ of letters in $\mathcal{A}$ such that $a_N = a_N'$ and for all $n \geq N$, $a_n'$ occurs in $\tau_n(a_{n+1}')$.
Again using the fact that $|\tau_{[N,n)}(a'_n)|$ goes to infinity, this shows that $a_N$ belongs to $\mathcal{L}(X_{\boldsymbol{\tau}}^{(N)})$.
We conclude that $(X_{\boldsymbol{\tau}}^{(N)},S)$ is not minimal, a contradiction.
\end{proof}
| 2,761 | 37,977 |
en
|
train
|
0.39.7
|
\subsection{Dendric subshifts} \label{subsec:tree}
We now describe a subclass of the family of primitive unimodular proper $\mathcal{S}$-adic subshifts, namely
the class of dendric subshifts, that encompasses Sturmian subshifts, Arnoux-Rauzy subshifts (see Section \ref{subsec:AR}), as well as subshifts generated by interval exchanges (see~\cite{BDFDLPRR:15}). The ternary words generated by the Cassaigne-Selmer multidimensional continued fraction algorithm also provide dendric subshifts~\cite{ArnouxLa:2018,CLL:17}.
Dendric subshifts are defined with respect to combinatorial properties of their language expressed in terms of extension graphs.
We recall the notion of dendric words and subshifts, studied in \cite{BDFDLPRR:15,BDFDLPRR:15bis,BFFLPR:2015,BDFDLPR:16,Rigidity}.
Let $(X,S)$ be a minimal subshift defined on the alphabet $\mathcal{A}$.
For $w \in {\mathcal L}_X$, let
$$
\begin{array}{lcl}
L(w) = \{ a \in {\mathcal A} \mid aw \in {\mathcal L}_X\}, & & \ell(w) = \Card(L(w)), \\
R(w) = \{ a \in {\mathcal A} \mid wa \in {\mathcal L}_X\}, & & r(w) = \Card(R(w)).
\end{array}
$$
A word $w \in \mathcal{L}_X$ is said to be {\em right special} (resp. {\em left special}) if $r(w)\geq 2$ (resp. $\ell(w)\geq 2$). It is {\em bispecial} if it is both left and right special.
For a word $w \in \mathcal{L}(X)$, we consider the undirected bipartite graph $\E(w)$ called its \emph{extension graph} with respect to $X$ and defined as follows:
its set of vertices is the disjoint union of $L(w)$ and $R(w)$ and its edges are the pairs $(a,b) \in L(w) \times R(w)$ such that $awb \in \mathcal{L}(X)$.
For an illustration, see Example~\ref{ex:fibo} below.
We then say that a subshift $X$ is a \emph{dendric subshift} if, for every word $w \in {\mathcal L}(X)$, the graph $\E(w)$ is a tree.
Note that the extension graph associated with every non-bispecial word is trivially a tree.
We will consider here only minimal dendric subshifts.
The factor complexity of a dendric subshift over a $d$-letter alphabet is $(d-1)n + 1$ (see~\cite{BDFDLPR:16}),
and on a two-letter alphabet, the minimal dendric subshifts are the Sturmian subshifts.
Thus minimal dendric subshift are aperiodic when $d$ is greater or equal to $2$.
\begin{example}\label{ex:fibo}
\rm
Let $\sigma$ be the Fibonacci substitution defined over the alphabet $\{a,b\}$ by $\sigma \colon a \mapsto ab, b \mapsto a$ and consider the subshift generated by $\sigma$ (i.e., the set of bi-infinite words over $\mathcal{A}$ whose factors
belong to some $\sigma^n (a)$).
The extension graphs of the empty word and of the two letters $a$ and $b$ are represented in Figure~\ref{fig:fibo-ext}.
\begin{figure}
\caption{The extension graphs of $\varepsilon$ (on the left), $a$ (on the center) and $b$ (on the right) are trees.}
\label{fig:fibo-ext}
\end{figure}
\end{example}
The following theorem states a structural property
for return words of minimal dendric subshifts, from which a description as primitive unimodular proper $\mathcal{S}$-adic subshifts can be deduced (Proposition~\ref{prop:DendricareSadic} below).
Let $(X,S)$ be a minimal subshift over the alphabet $\mathcal{A}$ and let $w \in \mathcal{L}(X)$.
A {\em return word} to $w$ is a word $v$ in $\mathcal{L}(X)$ such that $w$ is a prefix of $vw$ and $vw$ contains exactly two occurrences of $w$.
We recall below a corollary of \cite[Theorem 4.5]{BDFDLPR:16}.
\begin{theorem}\label{theo:return}
Let $(X,S)$ be a minimal dendric subshift defined on the alphabet $\mathcal{A}$. Then, for any $w \in \mathcal{L}(X)$, the set of return words to $w$ is a basis of the free group on $\mathcal{A}$.
\end{theorem}
In particular, dendric subshifts have bounded topological rank.
The next result shows that minimal dendric subshifts are primitive unimodular proper $\mathcal{S}$-adic subshifts.
Similar results are proved with the same method in~\cite{BFFLPR:2015,Rigidity,BSTY} but not highlighting all the properties stated below, so we provide a proof for the sake of self-containedness.
It relies on $\mathcal{S}$-adic representations built from return words \cite{Durand:2000,Durand:03} together with the remarkable property of return words of dendric subshifts stated in Theorem \ref{theo:return}.
We also provide in
Section \ref{subsection:vs} an example of a primitive unimodular proper subshift which is not dendric and whose strong orbit equivalence class contains no dendric subshift.
\begin{proposition}\label{prop:DendricareSadic}
Minimal dendric subshifts are primitive unimodular proper $\mathcal{S}$-adic subshifts.
\end{proposition}
\begin{proof}
Let $(X,S)$ be a minimal dendric subshift over the alphabet ${\mathcal A}= \{1,2, \dots,d\}$ and take any $x\in X$. For every $n\geq 1$, let $V_n(x):=\{v_{1,n},\cdots,v_{d,n}\}$ be the set of return words to $x_{[0,n)}$ and $V_0 (x) = {\mathcal A}$.
We stress the fact that $V_n(x)$ has cardinality $d$ for all $n$, according to Theorem~\ref{theo:return}.
Let $(n_i)_{i \geq 1}$ be a strictly increasing integer sequence such that $n_1 = 1$ and such that each $v_{j,n_i}x_{[0,n_i)}$ occurs in $x_{[0,n_{i+1})}$ and in each $v_{k,n_{i+1}}$.
Let $\theta_i $ be an endomorphism of ${\mathcal A}^*$ such that $\theta_i ({\mathcal A}) = V_{n_i} (x)$.
Since $x_{[0,n_i)}$ is a prefix of $x_{[0,n_{i+1})}$, any $v_{j,n_{i+1}}\in V_{n_{i+1}}(x)$ has a unique decomposition as a concatenation of elements $v_{k,n_i}\in V_{n_i}(x)$.
More precisely, for any $v_{j,n_{i+1}}\in V_{n_{i+1}}(x)$, there is a unique sequence $(v_{k_j(1),n_i},v_{k_j(2),n_i},\dots,v_{k_j(\ell_j),n_i})$ of elements of $V_{n_i}(x)$ such that $v_{k_j(1),n_i} \cdots v_{k_j(\ell_j),n_i} = v_{j,n_{i+1}}$ and for all $m \in \{1,\dots,\ell_j\}$, $v_{k_j(1),n_i} \cdots v_{k_j(m),n_i} x_{[0,n_i)}$ is a prefix of $v_{j,n_{i+1}} x_{[0,n_{i+1})}$.
This induces a unique endomorphism $\lambda_i$ of ${\mathcal A}^*$ defined by $\theta_{i+1} = \theta_i \circ \lambda_i$.
From the choice of the sequence $(n_i)_{i\geq 1}$, the matrices $M_{\lambda_i}$ have positive coefficients, so the sequence of morphisms $(\lambda_i)_{i \geq 1}$ is primitive.
Furthermore, as $x_{[0,n_{i+1}]}$ is prefix of each $v_{j,n_{i+1}}x_{[0,n_{i+1)}}$, there exists some $v \in V_{n_i}(x)$ such that $v_{k_j(1)}=v$ for all $j$.
In other words, the morphisms $\lambda_i$ are left proper.
Finally, from Theorem~\ref{theo:return}, the matrices $M_{\lambda_i}$ are unimodular.
Hence $(X,S)$ is $\mathcal{S}$-adic generated by the primitive directive sequence of unimodular left proper endomorphisms $\boldsymbol{\lambda} = (\lambda_i)_{i\geq 1}$.
We deduce from Lemma~\ref{lemma:proper} that minimal dendric subshifts are primitive unimodular proper $\mathcal{S}$-adic subshifts.
\end{proof}
Observe that using Lemma~\ref{lemma:aperiodicity} we recover that minimal dendric subshifts on at least two letters are aperiodic.
| 2,482 | 37,977 |
en
|
train
|
0.39.8
|
\section{Dimension groups of primitive unimodular proper $\mathcal{S}$-adic subshifts} \label{sec:proofs2}
In this section we first prove a key result of this paper, namely Theorem \ref{theo:cohoword}, which states that $H(X,T)= C (X, \mathbb{Z})/ \beta C (X, \mathbb{Z})$ is generated as an additive group by the classes of the characteristic functions of letter cylinders.
We then deduce a simple expression for the dimension group of primitive unimodular proper $\mathcal{S}$-adic subshifts.
\subsection{From letters to factors}
We recall that $\chi_U$ stands for the characteristic function of the set $U$.
\begin{theorem}
\label{theo:cohoword}
Let $(X,S)$ be a primitive unimodular proper $\mathcal{S}$-adic subshift.
Any function $f\in C (X, \mathbb{Z})$ is cohomologuous to some integer linear combination of the form $\sum_{a\in \mathcal{A}} \alpha_a \chi_{[a]}\in C (X,\mathbb{Z})$. Moreover, the classes $[\chi_{[a]}]$, $a\in \mathcal{A}$, are $\mathbb{Q}$-independent.
\end{theorem}
\begin{proof}
Let $\boldsymbol{\tau} = (\tau_n:\mathcal{A}^* \to \mathcal{A}^*)_{n\geq 1}$ be a primitive unimodular proper directive sequence of $(X,S)$, hence $X = X_{\boldsymbol{\tau}}$.
Using Proposition~\ref{prop:minimalSSIprimitif}, all subshifts $(X_{\boldsymbol{\tau}}^{(n)},S)$ are minimal and aperiodic and $\min_{a \in \mathcal{A}} |\tau_{[1,n)}(a)|$ goes to infinity when $n$ increases.
Let us show that the group $H(X,S )=C(X, {\mathbb Z})/ \beta C(X,{\mathbb Z}) $ is spanned by the set of classes of characteristic functions of letter
cylinders
$\{ [\chi_{[a]}] \mid a \in \mathcal{A} \}$.
From Theorem~\ref{theo:BSTY} and using the fact that $(X,S)$ is minimal and aperiodic, one has, for all positive integer $n$, that
$$
\mathcal{P}_n = \{ S^k \tau_{[1,n]} ([a]) \mid 0\leq k < |\tau_{[1,n]} (a)| , a \in \mathcal{A} \}
$$
is a finite partition of $X$ into clopen sets. This provides a family of nested Kakutani-Rohlin tower partitions.
The morphisms of the directive sequence $\boldsymbol{\tau}$ being proper, for all $n$, there are letters $a_n, b_n$ such that all images $\tau_n (c)$, $c \in \mathcal{A}$, start with $a_n$ and end with $b_n$.
From this, it is classical to check that $(\mathcal{P}_n)_n$ generates the topology of $X$ (the proof is the same as~\cite[Proposition 14]{DHS:99} that is concerned with the particular case $\tau_{n+1}=\tau_n$ for all $n$).
We first claim that $H(X,S ) $ is spanned by the set of classes $\cup_n \Omega_n $, where
$$\Omega_n = \{ [\chi_{\tau_{[1,n]} ([a])}] \mid a \in \mathcal{A} \} \quad n \geq 1.$$
In other words, $H(X,S)$ is spanned by the set of classes of characteristic functions of bases of the sequence of partitions $({\mathcal P})_n$.
It suffices to check that, for all $u^- u^+\in \mathcal{L} (X)$, the class $[\chi_{[u^-.u^+]}]$ is a linear integer combination of elements belonging
to some $\Omega_n$.
Let us check this assertion. Let $u^-u^+$ belong to $\mathcal{L} (X)$.
Since $\min_{a \in \mathcal{A}} |\tau_{[1,n)}(a)|$ goes to infinity, there exists $n$ such that $|u^-|,|u^+| < \min_{a\in \mathcal{A}} |\tau_{[1,n)} (a) |$.
The directive sequence $ \boldsymbol{\tau}$ being proper, there exist words $w,w'$ with respective lengths $|w|=|u^-|$ and $|w'|=|u^+|$
such that all images $\tau_{[1,n]} (a)$ start with $w$ and end with $w'$.
Let $x \in [u^-. u^+]$.
Let $a \in \mathcal{A}$ and $k \in \mathbb{N}$, $0\leq k < |\tau_{[1,n]} (a)|$, such that $x$ belongs to the atom $S^k \tau_{[1,n]} ([a])$.
Observing that $\tau_{[1,n]} ([a])$ is included in $[w'.\tau_{[1,n]} (a)w]$, this implies that the full atom $S^k \tau_{[1,n]} ([a])$ is included in $[u^-. u^+]$.
Consequently $[u^-. u^+]$ is a finite union of atoms in $\mathcal{P}_n$.
But each characteristic function of an atom of the form $ S^k \tau_{[1,n]} ([a]) $ is cohomologous to $\chi_{\tau_{[1,n]} ([a])}$. The proof works as in the proof of Proposition \ref{prop:cob}. This thus proves the claim.
Now we claim that each element of $\Omega_n $ is a linear integer combination of elements in $\{ [\chi_{[a]}] \mid a \in \mathcal{A} \}$.
More precisely, let us show that $\chi_{\tau_{[1,n]}([b])}$ is cohomologous to
$$
\sum_{a\in \mathcal{A}} (M_{\tau_{[1,n]}}^{-1})_{b,a}\chi_{[a]} .
$$
Let $a\in \mathcal{A}$ and $n\geq 1$.
One has $[a] = \cup_{B \in \mathcal{P}_n } (B \cap [a]) $ and thus $\chi_{[a]} $ is cohomologous to the map
$$
\sum_{b\in \mathcal{A}} (M_{\tau_{[1,n]}})_{a,b}\chi_{\tau_{[1,n]}([b])} ,
$$
by using the fact that the maps $ \chi_{S^k \tau_{[1,n]} ([a]) }$ are cohomologous to $\chi_{\tau_{[1,n]} ([a])}$.
This means that for $U = ([\chi_{[a]} ] ) _{a\in \mathcal{A}} \in H (X,S)^\mathcal{A}$ and $V = ([\chi_{\tau_{[1,n]}([a])}])_{a\in \mathcal{A}} \in H (X,S)^\mathcal{A}$, one has
$$
U = M_{\tau_{[1,n]}}V
$$
and as a consequence $V = M_{\tau_{[1,n]}}^{-1}U$.
This proves the claim and the first part of the theorem.
To show the independence, suppose that there exists some row vector $\alpha = (\alpha_a )_{a\in \mathcal{A}} \in \mathbb{Z}^\mathcal{A}$ such that $\sum_a \alpha_a [\chi_{[a]}] = 0$.
Hence there is some $f\in C (X, \mathbb{Z} )$ such that $\sum_a \alpha_a \chi_{[a]} = f\circ S - f$.
We now fix some $n$ for which $f$ is constant on each atom of $\mathcal{P}_n$.
Observe that for all $x\in X$ and all $k \in \mathbb{N}$, one has $f (S^k x ) - f(x) = \sum_{j=0}^{k-1} \alpha_{x_j}$.
Let $c\in \mathcal{A}$ and $x\in \tau_{[1,n+1]} ([c])$.
Then, $x$ and $S^{ |\tau_{[1,n+1]} (c)|} (x)$ belong to $\tau_{[1,n]} ([a_{n+1}])$.
Hence, $f (S^{ |\tau_{[1,n+1]} (c)|} x ) - f(x) = 0 $, and thus
$$
(\alpha M_{\tau_{[1,n]}})_c = \sum_{j=0}^{|\tau_{[1,n+1]} (c)|-1} \alpha_{x_j} = 0 .
$$
This holds for all $c$, hence $\alpha M_{\tau_{[1,n]}} = 0$, which yields $\alpha = 0$, by invertibility of the matrix $ M_{\tau_{[1,n]}}$.
\end{proof}
Observe that in the previous result, we can relax the assumption of minimality. Indeed,
one checks that the same proof works if we assume that $(X,S)$ is aperiodic (recognizability then holds by \cite{BSTY})
and that $\min_{a \in \mathcal{A}} |\tau_{[1,n)}(a)|$ goes to infinity.
We now derive two corollaries from Theorem~\ref{theo:cohoword} dealing respectively with invariant measures and with the image subgroup.
Note that Corollary \ref{coro:measures} extends a statement initially proved for interval exchanges \cite{FerZam:08}.
See also \cite{BHL:2019} for a similar result in the framework of automorphisms of the free group and \cite{BHL:2020} for subshifts with finite rank.
\begin{corollary}\label{coro:measures}
Let $(X,S)$ be a primitive unimodular proper $\mathcal{S}$-adic subshift over the alphabet $\mathcal{A}$ and let $\mu, \mu' \in \mathcal M (X,S)$.
If $\mu$ and $\mu'$ coincide on the letters, then they are equal,
that is, if $\mu([a]) = \mu'([a])$ for all a in $\mathcal{A}$, then $\mu(U) = \mu'(U)$, for any clopen subset $U$ of $X$.
\end{corollary}
\begin{corollary}\label{coro:freq}
Let $(X,S)$ be a primitive unimodular proper $\mathcal{S}$-adic subshift over the alphabet ${\mathcal A}$.
The image subgroup of $(X,S)$ satisfies
\begin{align*}
I(X,S) & = \bigcap_{\mu\in\mathcal{M}(X,S)} \langle {\mu([a]) : a\in A} \rangle \\
& = \left\lbrace \alpha : \exists (\alpha_a)_{a\in \mathcal{A}} \in \mathbb{Z}^\mathcal{A} , \alpha
= \sum_{a\in\mathcal{A}} \alpha_a \mu([a]) \ \forall \mu \in \mathcal{M} (X,S) \right\rbrace .
\end{align*}
\end{corollary}
The proof of Corollary \ref{coro:freq} uses the two descriptions of the image subgroup given in Proposition \ref{prop:GW} and Proposition \ref{prop:cob}.
In both corollaries, the assumption of being proper can be dropped. The proof then uses the measure-theoretical Bratteli-Vershik representation of the primitive unimodular $\mathcal{S}$-adic subshift given in \cite[Theorem 6.5]{BSTY}.
| 2,993 | 37,977 |
en
|
train
|
0.39.9
|
\subsection{An explicit description of the dimension group}
Theorem~\ref{theo:cohoword} allows a precise description of the dimension group of primitive unimodular proper $\mathcal{S}$-adic subshifts.
Note that in the case of interval exchanges, one recovers the results obtained in \cite{Putnam:89}; see also \cite{Putnam:92,GjerdeJo:02}.
We first need the following Gottschalk-Hedlund type statement \cite{GotHed:55}.
\begin{lemma}[\cite{Host:95}, Lemma 2, \cite{DHP}, Theorem 4.2.3]
\label{lemma:positivecohomologous}
Let $(X,T)$ be a minimal Cantor system and let $f \in C(X,\mathbb{Z})$.
There exists $g \in C(X,\mathbb{N})$ that is cohomologous to $f$ if and only if for every $x \in X$, the sequence $\left( \sum_{k=0}^n f \circ T^k(x)\right)_{n \geq 0}$ is bounded from below.
\end{lemma}
We now can deduce one of our main statements.
\begin{theorem}\label{theo:dg}
Let $(X,S)$ be a primitive unimodular proper $\mathcal{S}$-adic subshift over a $d$-letter alphabet.
The linear map $\Phi : H (X,S) \to \mathbb{Z}^d$ defined by $\Phi ( [\chi_{[a]}] ) = e_a$, where $\{ e_a \mid a \in \mathcal{A} \}$ is the canonical base of $ \mathbb{Z}^d$,
defines an isomorphism of dimension groups from $K^{0}(X,S)$ onto
\begin{align}
\label{align:ZdGD}
\left( {\mathbb Z} ^d, \, \{ {\bf x} \in {\mathbb Z}^d \mid \langle {\bf x}, \boldsymbol{ \mu} \rangle > 0 \mbox{ for all }
\mu \in {\mathcal M} (X,S) \}\cup \{{\mathbf 0}\},\, \bf{1}\right ),
\end{align}
where the entries of $\bf{1}$ are equal to $1$.
\end{theorem}
\begin{proof}
From Theorem \ref{theo:cohoword}, $\Phi$ is well defined and is a group isomorphism from $H (X,S)$ onto $\mathbb{Z}^d$.
We obviously have $\Phi ( [1] ) = \Phi(\sum_{a \in \mathcal{A}} [\chi_{[a]}]) = \bf{1}$ and it remains to show that
\[
\Phi(H^+(X,S))
=
\{ {\bf x} \in {\mathbb Z}^d \mid \langle {\bf x}, \boldsymbol{ \mu} \rangle > 0 \mbox{ for all }
\mu \in {\mathcal M} (X,S) \}
\cup
\{{\mathbf 0}\}.
\]
Any element of $H^+(X,S)$ is of the form $[f]$ for some $f \in C(X,\mathbb{N})$.
From Theorem \ref{theo:cohoword}, there exists a unique vector ${\bf x} = (x_a)_{a \in \mathcal{A}}$ such that $[f] = \sum_{a \in \mathcal{A}} x_a [\chi_{[a]}]$.
As $f$ is non-negative, we have, for any $\mu \in {\mathcal M} (X,S)$,
\[
\langle \Phi([f]),\boldsymbol{\mu} \rangle
=
\sum_{a \in \mathcal{A}} x_a \mu([a])
=
\int f d\mu
\geq 0,
\]
with equality if and only if $f=0$ (in which case ${\bf x} = {\bf 0}$).
For the other inclusion, assume that ${\bf x} = (x_a)_{a \in \mathcal{A}} \in {\mathbb Z}^d$ satisfies $\langle {\bf x}, \boldsymbol{ \mu} \rangle > 0 \mbox{ for all }
\mu \in {\mathcal M} (X,S)$ (the case ${\bf x} = {\bf 0}$ is trivial).
We consider the function $f = \sum_{a \in \mathcal{A}} x_a \chi_{[a]}$. According to Lemma~\ref{lemma:positivecohomologous}, the existence of $f' \in [f]$ such that $f'$ is non-negative is equivalent to the existence of a lower bound for ergodic sums.
Assume by contradiction that there exists a point $x \in X$ such that the sequence $\left( \sum_{k=0}^n f \circ S^k(x)\right)_{n \geq 0}$ is not bounded from below.
Thus there is a an increasing sequence of positive integers $(n_i)_{i \geq 0}$ such that
\[
\lim_{i \to +\infty} \sum_{k=0}^{n_i-1} f \circ S^k(x) = -\infty.
\]
Passing to a subsequence $(m_i)_{i \geq 0}$ of $(n_i)_{i \geq 0}$ if necessary, there exists $\mu \in \mathcal{M}(X,S)$ satisfying
\[
\langle {\bf x}, \boldsymbol{\mu} \rangle = \int f d\mu = \lim_{i \to +\infty} \frac{1}{m_i} \sum_{k=0}^{m_i-1} f \circ S^k(x) \leq 0,
\]
which contradicts our hypothesis.
The sequence $\left( \sum_{k=0}^n f \circ S^k(x)\right)_{n \geq 0}$ is thus bounded from below and we conclude by using Lemma~\ref{lemma:positivecohomologous}.
\end{proof}
\begin{remark}
We cannot remove the hypothesis of being left or right proper in Theorem \ref{theo:dg}.
Consider indeed the subshift $(X,S)$ defined by the primitive unimodular non-proper substitution $\tau$ defined over $\{a,b\}^*$ as $\tau \colon a \mapsto aab, b \mapsto ba$.
According to \cite[p.114] {DurandThese}, the dimension group of $(X,S)$ is isomorphic to
$\left( {\mathbb Z}^3 , \left\{ {\mathbf x} \in {\mathbb Z}^3 : \langle {\mathbf x}, {\mathbf v} \rangle > 0 \right\} , (2,0,-1) \right)$
where ${\mathbf v}= ( \frac{1+ \sqrt 5}{2}, 2,1)$.
\end{remark}
\subsection{Ergodic measures}\label{subsection:ergodic}
We now focus on further consequences of Theorem \ref{theo:cohoword} for invariant measures of primitive unimodular proper $\mathcal{S}$-adic subshifts.
\begin{corollary}\label{cor:oe}
Two primitive unimodular proper $\mathcal{S}$-adic subshifts $(X_1,S)$ and $(X_2,S)$ are strong orbit equivalent
if and only if
there is a unimodular matrix $M$ such that $M {\bf 1} = {\bf 1}$ and
\[
\{\boldsymbol{\nu} \mid \nu \in \mathcal M(X_2,S)\}
=
\{M^{\rm T} \boldsymbol{\mu} \mid \mu \in \mathcal M(X_1,S)\}.
\]
In particular, $(X_1,S)$ and $(X_2,S)$ are defined on alphabets with the same cardinality.
\end{corollary}
\begin{proof}
For $i = 1,2$, let $\Phi_i:H(X_i,S) \to \mathbb{Z}^{d_i}$ be the map given in Theorem~\ref{theo:dg}, where $d_i$ is the cardinality of the alphabets $\mathcal{A}_i$ of $X_i$.
Let us also write ${\bf 1}_i$ the vector of dimension $d_i$ only consisting in $1$'s and
\[
C_i
=
\Phi_i(H^+(X_i,S))
=
\{ {\bf x} \in {\mathbb Z}^{d_i}
\mid
\langle {\bf x}, \boldsymbol{ \mu} \rangle > 0
\mbox{ for all } \mu \in {\mathcal M} (X_i,S) \}
\cup \{{\mathbf 0}\},
\]
so that $\Phi_i$ defines an isomorphism of dimension groups from $K^0(X_i,S)$ onto $(\mathbb{Z}^{d_i},C_i,{\bf{1}}_i)$.
First assume that $(X_1,S)$ and $(X_2,S)$ are strong orbit equivalent.
Theorem~\ref{oe} implies that there is an isomorphism of dimension group from $(\mathbb{Z}^{d_2},C_2,{\bf{1}}_2)$ onto $(\mathbb{Z}^{d_1},C_1,{\bf{1}}_1)$.
Hence $d_1 = d_2 = d$ and this isomorphism is given by a unimodular matrix $M$ of dimension $d$ satisfying $M{\bf 1} = {\bf 1}$ (where ${\bf 1} = {\bf 1}_1 = {\bf 1}_2$) and $MC_2 = C_1$.
We also denote by $M$ the map ${\bf x} \in \mathbb{Z}^d \mapsto M {\bf x}$.
Recall from Section~\ref{subsec:cyl} that the map
\[
\mu \in \mathcal M(X_i,S) \mapsto \left(\tau_\mu: [f] \in H(X_i,S) \mapsto \int f d\mu\right)
\]
is an affine isomorphism from $\mathcal M(X_i,S)$ to $\T(K^0(X_i,S))$.
Observing that for all $\mu \in \mathcal M(X_1,S)$, $\tau_\mu \circ \Phi_1^{-1} \circ M \circ \Phi_2$ is a trace of $K^0(X_2,S)$, it defines an affine isomorphism
$\mu \in \mathcal M(X_1,S) \mapsto \nu \in \mathcal M(X_2,S)$, where $\nu$ is such that $\tau_\nu = \tau_\mu \circ \Phi_1^{-1} \circ M \circ \Phi_2$.
Since $\boldsymbol{\mu} = (\tau_\mu([\chi_{[a]}]))_{a \in \mathcal{A}_1}$, we have, for all $a \in \mathcal{A}_2$,
\[
\nu([a])
= \tau_\nu( [\chi_{[a]}])
= \tau_\mu \circ \Phi_1^{-1} \circ M \circ \Phi_2 ( [\chi_{[a]}])
= \boldsymbol{\mu}^{\rm T} M e_a
= e_a^{\rm T} M^{\rm T} \boldsymbol{\mu},
\]
so that $\boldsymbol{\nu} = M^{\rm T} \boldsymbol{\mu}$.
Now assume that we are given a unimodular matrix $M$ satisfying $M {\bf 1} = {\bf 1}$ and
\[
\{\boldsymbol{\nu} \mid \nu \in \mathcal M(X_2,S)\}
=
\{M^{\rm T} \boldsymbol{\mu} \mid \mu \in \mathcal M(X_1,S)\}.
\]
In particular, this implies that $d_1 = d_2=d$.
Let us show that the map $M: {\bf x} \in \mathbb{Z}^d \mapsto M {\bf x}$ defines an isomorphism of dimension groups from $(\mathbb{Z}^{d},C_1,{\bf{1}})$ to $(\mathbb{Z}^{d},C_2,{\bf{1}})$.
We only need to show that $MC_1 = C_2$.
The matrix $M$ being unimodular, we have $M {\bf x} = {\bf 0}$ if and only if ${\bf x} = {\bf 0}$.
For ${\bf x} \neq {\bf 0}$, we have
\begin{align*}
{\bf x} \in C_2
& \Leftrightarrow \langle {\bf x},\boldsymbol{\nu} \rangle >0 \text{ for all } \nu \in \mathcal{M}(X_2,S) \\
& \Leftrightarrow \langle {\bf x},M^{\rm T}\boldsymbol{\mu} \rangle >0 \text{ for all } \mu \in \mathcal{M}(X_1,S) \\
& \Leftrightarrow \langle M {\bf x},\boldsymbol{\mu} \rangle >0 \text{ for all } \mu \in \mathcal{M}(X_1,S) \\
& \Leftrightarrow M {\bf x} \in C_1,
\end{align*}
which ends the proof.
\end{proof}
According to Theorem \ref{theo:dg}, dimension groups of primitive unimodular proper subshifts have rank $d$.
This implies that the number $e$ of ergodic measures satisfies $e\leq d$.
In fact, we have even more from the following result.
\begin{proposition}\label{prop:effrosshen81}
\cite[Proposition 2.4]{Effros&Shen:1981}
Finitely generated simple dimension groups of rank $d$ have at most $d-1$ pure traces.
\end{proposition}
Dimension groups of minimal Cantor systems $(X,T)$ are simple dimension groups (Theorem \ref{theo:simpleminimal}) and, since the Choquet simplex of traces is affinely isomorphic to the simplex of ergodic measures, we derive the following.
\begin{corollary}\label{cor:d-1}
Primitive unimodular proper $\mathcal{S}$-adic subshifts over a $d$-letter alphabet have at most $d-1$ ergodic measures.
\end{corollary}
If the primitive unimodular proper $\mathcal{S}$-adic subshift $(X,S)$ has some extra combinatorial properties, then the number of ergodic measures can be smaller.
Suppose indeed that $(X,S)$ is a minimal dendric subshift on a $d$-letter alphabet.
As its factor complexity equals $(d-1)n + 1$, one has a priori $e \leq d-2$ for $d \geq 3$ according to \cite[Theorem 7.3.4]{CANT}.
One can even have more as a direct consequence of \cite{Damron:2019} and \cite{DolPer:2019}. Note that this statement encompasses the case of interval exchanges handled in \cite{Katok:73,Veech:78}.
\begin{theorem} \label{theo:ergodic}
Let $(X,S)$ be a minimal dendric subshift over a $d$-letter alphabet.
One has
$$\mbox{\rm Card}({\mathcal M}_e (X,S)) \leq \frac{d}{2}.$$
\end{theorem}
\begin{proof}
According to \cite{Damron:2019}, a minimal subshift is said to satisfy the
regular bispecial condition if any large enough bispecial word $w$ has only one left extension $aw \in \mathcal{L}(X)$, $a \in \mathcal{A}$, that is right special and only one right extension $wa \in \mathcal{L}(X)$, $a \in \mathcal{A}$, that is left special. Now we use the fact that minimal dendric subshifts satisfy the regular bispecial condition according to \cite{DolPer:2019}.
We conclude by using the upper bound on the number ergodic measures from \cite{Damron:2019}.
\end{proof}
| 4,015 | 37,977 |
en
|
train
|
0.39.10
|
\section{Infinitesimals and balance property} \label{sec:saturation}
When the infinitesimal subgroup $ \mbox{\rm Inf} (K^0 (X,T))$ of a minimal Cantor system $(X,T)$ is trivial, the system is called {\em saturated}. This property is proved in \cite{TetRob:2016} to hold for primitive, aperiodic, irreducible substitutions for which images of letters have a common prefix.
At the opposite, an example of a dendric subshift with non-trivial infinitesimal subgroup is provided in Example \ref{ex:nontrivial}.
A formulation of saturation in terms of the topological full group is given in \cite{BezKwia:00}. Recall also that for saturated systems, the quotient group $I(X,T)/E(X,T)$ is torsion-free by \cite[Theorem 1]{CortDP:16} (see also \cite{GHH:18}).
We first state a characterization of the triviality of the infinitesimal subgroup $\mbox{\rm Inf} (K^0 (X,S))$ for minimal unimodular proper $\mathcal{S}$-adic subshifts (see Proposition \ref{theo:saturated}). We then relate the saturation property with a combinatorial notion called {\em balance property} and we provide a topological characterization of primitive
unimodular proper $\mathcal{S}$-adic subshifts that are balanced (see Corollary \ref{cor:balanced}).
\begin{proposition}\label{theo:saturated}
Let $(X,S)$ be a minimal unimodular proper $\mathcal{S}$-adic subshift on a $d$-letter alphabet $A$.
The infinitesimal subgroup $\mbox{Inf} (K^0 (X,S))$ is non-trivial
if and only if there is a non-zero vector ${\mathbf x} \in {\mathbb Z}^d$
orthogonal to any element of the simplex of letter measures.
\noindent In particular, if there exists some invariant measure $\mu \in {\mathcal M}(X,S)$ for which the frequencies of letters $\mu([a])$, $a \in A$, are rationally independent, then the infinitesimal subgroup $\mbox{Inf} (K^0 (X,S))$ is trivial.
\end{proposition}
\begin{proof}
According to Theorem \ref{theo:dg}, the elements of $ \mbox{\rm Inf} (K^0 (X,S))$ are the classes of functions that are represented by vectors ${\mathbf x} \in {\mathbb Z}^d$ such that
$\langle {\mathbf x}, \boldsymbol{ \mu} \rangle =0$ for every $\mu \in {\mathcal M} (X,S) $.
Recall also that coboundaries are represented by the vector ${\mathbf 0}$.
Hence $\mbox{Inf} (K^0 (X,S))$ is not trivial if and only if there exists $ {\mathbf x} \in {\mathbb Z}^d$, with ${\mathbf x} \neq {\mathbf 0}$,
such that $\langle {\bf x}, \boldsymbol{ \mu} \rangle = 0$, for every $\mu \in \mathcal{M}(X,S)$.
Assume now that there exists some invariant measure $\mu \in {\mathcal M}(X,S)$ for which the frequencies of letters are rationally independent.
Hence, for any vector ${\mathbf x} \in {\mathbb Z}^d$, $\langle {\bf x}, \boldsymbol{ \mu} \rangle = 0$ implies that ${\mathbf x}={\mathbf 0}$.
From above, this implies that $\mbox{Inf} (K^0 (X,S))$ is trivial.
\end{proof}
See Example \ref{ex:nontrivial} for an example of a dendric subshift with non-trivial infinitesimals.
We now introduce a notion of balance for functions. Let $(X,T)$ be a minimal Cantor system.
We say that $f \in C (X , \mathbb{R} )$ is {\em balanced} for $(X,T)$ whenever there exists a constant $C_f>0$
such that $$|\sum_{i=0}^n f(T^ix) - f(T^iy) | \leq C_f \mbox { for all }x, y \in X \mbox{ and for all } n .$$
Balance property is usually expressed for letters and factors (see for instance
\cite{BerCecchi:2018}). Indeed a minimal subshift $(X,S)$ is said to be {\em balanced on the factor} $v \in {\mathcal L} (X)$ if $\chi_{[v]}: X \to \{0,1\}$ is balanced, or, equivalently,
if there exists a constant $C_v$ such that for all $w,w'$ in $\mathcal{L}_X$ with $|w|=|w'|$, then $||w|_v-|w'|_v|\leq C_v.$
It is {\em balanced on letters} if it is balanced on each letter, and it is {\em balanced on factors} if it is balanced on all its factors.
More generally, we say that a system $(X,T)$ is balanced on a subset $H \subset C(X, \mathbb{R})$ whenever it is balanced for all $f$ in $H$.
It is standard to check that any system $(X,T)$ is balanced on the (real) coboundaries.
Of course, a subshift $(X,S)$ is balanced on a generating set of $C(X,\mathbb{Z})$ if and only if it is balanced on factors or, equivalently,
if every $f\in C(X,\mathbb{Z})$ is balanced.
One can observe that the balance property on letters is not necessarily preserved under topological conjugacy whereas the balance property on factors is.
Indeed, consider the shift generated by the Thue--Morse substitution $\sigma \colon a \mapsto ab, b \mapsto ba$.
It is clearly balanced on letters.
It is conjugate to the shift generated by the substitution
$\tau \colon a \mapsto bb$, $b\mapsto bd$, $c \mapsto ca$, $d\mapsto cb$ via the sliding block code
$00 \mapsto a,$ $01 \mapsto b,$ $10 \mapsto c,$ $11 \mapsto d$ (see \cite[p.149]{Queffelec:2010}).
The subshift generated by $\tau$ is not balanced on letters (see \cite{BerCecchi:2018}).
Next proposition will be useful to characterize balanced functions of a system $(X,T)$.
\begin{proposition}\label{prop:carBalanced}
Let $(X,T)$ be a minimal dynamical system. An integer valued continuous function $f \in C(X, {\mathbb Z})$ is balanced for $(X,T)$ if and only if there exists $\alpha \in {\mathbb R}$
such that the map $f-\alpha$ is a real coboundary.
In this case, $\alpha = \int f d\mu$, for any $T$-invariant probability measure $\mu$ in $X$.
\end{proposition}
\begin{proof}
If the function $f-\alpha$ is a real coboundary, one easily checks that $f$ is balanced.
Moreover, the integral with respect to any $T$-invariant probability measure is zero, providing the last claim.
Assume that $f \in C(X, {\mathbb R})$ is balanced for $(X,T)$. Let $C>0$ be a constant such that
$ |\sum_{i=0}^n f\circ T^{i}(x)-f\circ T^{i}(y)| \le C$ holds uniformly in $x,y \in X$ for all $n\ge 0$.
Thus, for any non-negative integer $p \in \mathbb{N}$, there exists $N_{p}$ such that, for any $x\in X$, one has the following inequalities:
$$N_{p} \le \sum_{i=0}^p f\circ T^i(x) \le N_{p} +C.$$
Moreover, one checks that, for any $p,q \in \mathbb{N}$:
$$
qN_{p} \le \sum_{i=0}^{pq} f\circ T^i(x) \le qN_{p} +qC \textrm{ and } pN_{q} \le \sum_{i=0}^{pq} f\circ T^i(x) \le pN_{q} +pC.
$$
It follows that $-qC \le qN_{p} -pN_{q} \le pC$ and thus $-C/p \le N_{p}/p -N_{q}/q \le C/q $. Hence the sequence $(N_{p}/p)_{p}$ is a Cauchy sequence.
Let $\alpha = \lim_{p\to \infty} N_{p}/p$. By letting $q$ going to infinity, we get $ -C \le N_{p}-p\alpha \le 0$, so that $-C \le \sum_{i=0}^p f\circ T^i(x) -p\alpha \le C$ for any $x\in X$. By the classical Gottschalk--Hedlund's Theorem \cite{GotHed:55}, the function $f-\alpha$ is a real coboundary.
\end{proof}
As a corollary, we deduce that a minimal Cantor system $(X,T)$ balanced on $C(X,{\mathbb Z})$ is uniquely ergodic.
It also follows that for a minimal subshift $(X,S)$ balanced on the factor $v$, the frequency $\mu_{v} \in {\mathbb R}_{+}$ of $v$ exists,
i.e.,
for any $x \in X$,
$\lim_{ n \rightarrow \infty} \frac{ | x_{-n} \cdots x_0 \cdots x_{n}| _v }{ 2n+1}= \mu_v$, and even, the quantity $\sup _{n \in \mathbb{N}} | | x_{-n} \cdots x_0 \cdots x_{n}| _v - (2n+1) \mu_v| $ is finite (see also \cite{BerTij:2002}).
Actually, integer-valued continuous functions that are balanced for a minimal Cantor system $(X,T)$ are related to the continuous eigenvalues of the system as illustrated by the following folklore lemma. We recall that $E(X,T)$ stands for the set of additive continuous eigenvalues.
\begin{lemma}\label{lem:BalancedEigenvalue}
Let $(X,T)$ be a minimal Cantor system and let $\mu$ be a $T$-invariant measure. If $f \in C(X,\mathbb{Z})$ is balanced for $(X,T)$, then $\int f d\mu $ belongs to $ E(X,T)$.
\end{lemma}
\begin{proof}
If $f \in C(X,\mathbb{Z})$ is balanced for $(X,T)$, then so is $-f$ and there exists $g \in C(X,\mathbb{R})$ such that
$ -f+ \int f d\mu = g \circ T-g$ (by Proposition \ref{prop:carBalanced}).
This yields $\exp({2i\pi g\circ T})= \exp ({2i\pi \int f d \mu }) \exp ({ 2i \pi g})$ by noticing that $\exp({- 2i\pi f (x)})=1$ for any $x\in X$.
Hence $\exp ({ 2i \pi g})$ is a continuous eigenfunction associated with the additive eigenvalue $\int f d \mu$.
\end{proof}
| 2,999 | 37,977 |
en
|
train
|
0.39.11
|
As a corollary, we deduce that a minimal Cantor system $(X,T)$ balanced on $C(X,{\mathbb Z})$ is uniquely ergodic.
It also follows that for a minimal subshift $(X,S)$ balanced on the factor $v$, the frequency $\mu_{v} \in {\mathbb R}_{+}$ of $v$ exists,
i.e.,
for any $x \in X$,
$\lim_{ n \rightarrow \infty} \frac{ | x_{-n} \cdots x_0 \cdots x_{n}| _v }{ 2n+1}= \mu_v$, and even, the quantity $\sup _{n \in \mathbb{N}} | | x_{-n} \cdots x_0 \cdots x_{n}| _v - (2n+1) \mu_v| $ is finite (see also \cite{BerTij:2002}).
Actually, integer-valued continuous functions that are balanced for a minimal Cantor system $(X,T)$ are related to the continuous eigenvalues of the system as illustrated by the following folklore lemma. We recall that $E(X,T)$ stands for the set of additive continuous eigenvalues.
\begin{lemma}\label{lem:BalancedEigenvalue}
Let $(X,T)$ be a minimal Cantor system and let $\mu$ be a $T$-invariant measure. If $f \in C(X,\mathbb{Z})$ is balanced for $(X,T)$, then $\int f d\mu $ belongs to $ E(X,T)$.
\end{lemma}
\begin{proof}
If $f \in C(X,\mathbb{Z})$ is balanced for $(X,T)$, then so is $-f$ and there exists $g \in C(X,\mathbb{R})$ such that
$ -f+ \int f d\mu = g \circ T-g$ (by Proposition \ref{prop:carBalanced}).
This yields $\exp({2i\pi g\circ T})= \exp ({2i\pi \int f d \mu }) \exp ({ 2i \pi g})$ by noticing that $\exp({- 2i\pi f (x)})=1$ for any $x\in X$.
Hence $\exp ({ 2i \pi g})$ is a continuous eigenfunction associated with the additive eigenvalue $\int f d \mu$.
\end{proof}
We first give a statement valid for any minimal Cantor system that will then be applied below to primitive unimodular proper $\mathcal{S}$-adic subshifts.
We recall from \cite[Theorem 3.2, Corollary 3.6]{GHH:18} that there exists a one-to-one homomorphism $\Theta $ from $I (X,T)$ to $K^0 (X,T)$ such that, for $\alpha \in (0,1)\cap E (X,T)$, $\Theta (\alpha ) = [\chi_{U_{\alpha}}]$ where $U_\alpha$ is a clopen set, sucht that $\mu (U_\alpha)=\alpha$ for every
invariant measure $\mu$, and $ \chi_{U_{\alpha}} -\mu (U_{\alpha}) $ is a real coboundary. Hence $\chi_{U_{\alpha}}$ is balanced for $(X,T)$ (by Proposition \ref{prop:carBalanced}).
\begin{proposition}
\label{prop:balanced}
Let $(X,T)$ be a minimal Cantor system.
The following are equivalent:
\begin{enumerate}
\item
\label{item:1}
$(X,T)$ is balanced on some $H\subset C(X,\mathbb{Z})$ and $\{ [h] : h \in H \}$ generates $K^0 (X,T)$,
\item
\label{item:2}
$(X,T)$ is balanced on $C( X, \mathbb{Z})$,
\item
\label{item:3}
$\Theta (E(X,T))$ generates $K^0 (X,T)$.
\end{enumerate}
In this case $(X,T)$ is uniquely ergodic, then $I(X,T) = E(X,T)$ and ${\rm Inf } (K^0(X,T))$ is trivial.
\end{proposition}
\begin{proof}
Let us prove that \eqref{item:1} implies \eqref{item:2}.
Let $f\in C (X,\mathbb{Z} )$.
One has $[f] = \sum_{i=1}^{n} z_i [h_i]$ for some integers $z_i$ and some functions $h_i\in H$.
Hence $f = g\circ T - g + \sum_{i=1}^{n} z_i h_i$ for some $g\in C(X,\mathbb{Z} )$ and $f$ is balanced.
Consequently, $(X,T)$ is balanced on $C(X,\mathbb{Z})$.
It is immediate that \eqref{item:3} implies \eqref{item:1}.
Let us show that \eqref{item:2} implies \eqref{item:3}. Unique ergodicity holds by Proposition~\ref{prop:carBalanced}.
Let $\mu$ be the unique shift invariant probability measure of $(X,T)$.
For any $f\in C( X,\mathbb{Z})$, there are clopen sets $U_i$ and integers $z_{i}$ such that $f = \sum_{i=1}^n z_i \chi_{U_i}$.
From Lemma \ref{lem:BalancedEigenvalue} the values $ \mu(U_{i})\in I(X,T)$ are additive continuous eigenvalues in $E(X,T)$.
We get $[\chi_{U_{i}}] = \Theta(\mu(U_{i}))$ and $[f] = \sum_{i=1}^n z_i \Theta (\mu(U_{i}) )$.
This shows the claim \eqref{item:3}.
Assume that one of the three equivalent conditions holds.
Let $\mu$ denote the unique shift invariant probability measure.
Then, any map $f - \int f d \mu $, with $f\in C (X,\mathbb{Z} )$, is a real coboundary (by Proposition \ref{prop:carBalanced}). Hence, since any integer valued continuous function that is a real coboundary is a coboundary (\cite[Proposition 4.1]{Ormes:00}), the infinitesimal subgroup $ \mbox{\rm Inf} (K^0 (X,T))$ is trivial.
Moreover, Lemma \ref{lem:BalancedEigenvalue} implies that $I(X,T) \subset E(X,T)$.
The reverse implication $E(X,T) \subset I(X,T)$ comes from \cite[Proposition 11]{CortDP:16}, see also \cite[Corollary 3.7]{GHH:18}.
\end{proof}
Observe that when $(X,S)$ is a minimal subshift, by taking $H$
to be the set of classes of characteristic functions of cylinder sets, the balance
property is equivalent to the algebraic condition \eqref{item:3} of Proposition \ref{prop:balanced}.
We now provide a topological proof of the fact that the balance property on letters implies the balance property on factors
for primitive unimodular proper $\mathcal{S}$-adic subshifts.
For minimal dendric subshifts, this was already proved in \cite[Theorem 1.1]{BerCecchi:2018} using a combinatorial proof.
\begin{corollary} \label{cor:balanced}
Let $(X,S)$ be a primitive unimodular proper $\mathcal{S}$-adic subshift on a $d$-letter alphabet.
The following are equivalent:
\begin{enumerate}
\item
\label{item:balancedmaps}
$(X,S)$ is balanced for all integer valued continuous maps in $C (X , \mathbb{Z} )$,
\item
\label{item:balancedfactors}
$(X,S)$ is balanced on factors,
\item
\label{item:balancedletters}
$(X,S)$ is balanced on letters,
\item
\label{item:balancedrank}
$\mbox{\rm rank} (E(X,S)) =d$,
\end{enumerate}
and in this case $(X,S)$ is uniquely ergodic, $I(X,S) = E(X,S)$ and $\mbox{\rm Inf} (X,S)$ is trivial.
\end{corollary}
\begin{proof}
The implications $\eqref{item:balancedmaps} \Rightarrow \eqref{item:balancedfactors}\Rightarrow \eqref{item:balancedletters}$ are immediate.
Let us prove the implication $\eqref{item:balancedletters} \Rightarrow \eqref{item:balancedrank}$.
We deduce from Theorem \ref{theo:cohoword}, by taking $H$
to be the set of classes of characteristic functions of cylinder sets, that the conditions of Proposition \ref{prop:balanced} hold.
We deduce from Proposition \ref{theo:saturated} that \eqref{item:balancedrank} holds.
It remains to prove the implication $\eqref{item:balancedrank} \Rightarrow \eqref{item:balancedmaps}$. Suppose that $E(X,S)$ has rank $d$.
Let $\alpha_1 , \dots , \alpha_d\in E(X,S)$ be rationally independent.
There is no restriction to assume that they are all in $(0,1)$.
Consider, for $i =1\cdots,d$, $\Theta (\alpha_i ) = [\chi_{U_{\alpha_i}}]$ where $U_{\alpha_{i}}$ is a clopen set such that $\mu (U_{\alpha_{i}} ) = \alpha_i$ for any $S$-invariant measure $\mu \in {\mathcal M}(X,S)$ and $\chi_{U_{\alpha_{i}}}$ is balanced for $(X,S)$.
The classes $\Theta (\alpha_i)$'s are rationally independent because
the image of $\Theta (\alpha_i)$ by any trace is $\alpha_{i}$ and these values are assumed to be rationally independent.
As $K^0 (X,S)$ has rank $d$, by Theorem \ref{theo:cohoword}, and since it has no torsion (as recalled in Section~\ref{subsec:dim}), any element $[f]\in K^0 (X,S)$ is a rational linear combination of the $\Theta (\alpha_i )$'s. By Proposition \ref{prop:balanced}, any $f\in C(X, \mathbb{Z} )$ is balanced for $(X,S)$.
\end{proof}
\begin{remark}We deduce that primitive unimodular proper $\mathcal{S}$-adic subshifts that are balanced on letters have the maximal continuous eigenvalue group property, as defined in \cite{Durand&Frank&Maass:2019}, i.e., $E(X,S)= I(X,S)$. This implies in particular that non-trivial additive eigenvalues are irrational.
Indeed, by Corollary \ref{cor:balanced}, $\mbox{Inf} (K^0 (X,S))$ is trivial and thus, by Proposition \ref{theo:saturated}, the frequencies of letters (for the unique
shift-invariant measure) are rationally independent, which yields that $I(X,S)$ and thus $E(X,S)$ contain no rational non-trivial elements.
More generally, the fact that non-trivial additive eigenvalues are irrational hold for minimal dendric subshifts (even without
the balance property) \cite{Rigidity}. Note also that the triviality of $\mbox{Inf} (K^0 (X,S))$ says nothing about the balance property (see Example \ref{ex:balance}),
but the existence of non-trivial infinitesimals indicates that some letter is not balanced.
Lastly, the Thue--Morse substitution $\sigma \colon a \mapsto ab, b \mapsto ba$ generates
a subshift that is balanced on letters but not on factors \cite{BerCecchi:2018}. This substitution is neither unimodular, nor proper.
\end{remark}
| 3,084 | 37,977 |
en
|
train
|
0.39.12
|
\section{Examples and observations }\label{sec:examples}
\subsection{Brun subshifts}\label{subsec:Brun}
We provide a family of primitive unimodular proper $\mathcal{S}$-adic subshifts which are not dendric.
We consider the set of endomorphisms $S_\mathrm{Br} = \{\beta_{ab} \mid a \in \mathcal{A},\, b \in \mathcal{A} \setminus \{a\}\}$ over $d$~letters defined by
\[
\beta_{ab}:\ b \mapsto ab,\ c\mapsto c\ \text{for}\ c \in \mathcal{A} \setminus \{ b\}.
\]
A subshift $(X,S)$ is a \emph{Brun subshift} if it is generated by a primitive directive sequence $\boldsymbol{\tau } = (\tau_n)_n \in S_\mathrm{Br}^\mathbb{N}$
such that for all $n$ the endomorphism $\tau_n\tau_{n+1}$ belongs to
\begin{align*}
\big\{\beta_{ab}\beta_{ab} \mid a \in \mathcal{A},\, b \in \mathcal{A} \setminus \{a\}\big\}
\cup
\big\{\beta_{ab}\beta_{bc} \mid a \in \mathcal{A},\, b \in \mathcal{A} \setminus \{a\},\, c \in \mathcal{A} \setminus \{b\}\big\}.
\end{align*}
Observe that primitiveness of $\boldsymbol{\tau }$ is equivalent to the fact that for each $a \in \mathcal{A}$ there is $b \in \mathcal{A}$ such that $\beta_{ab}$ occurs infinitely often in $\boldsymbol{\tau }$.
Brun subshifts are not dendric in general: on a three-letter alphabet, they may contain strong and weak bispecial factors, hence that have an extension graph which is not a tree~\cite{Labbe&Leroy:16}.
However, we show below that they are primitive unimodular proper $\mathcal{S}$-adic subshifts.
\begin{lemma}
\label{lemma:brunproper}
Let $\mathcal{A}$ be a finite alphabet and $\gamma_{ab} : \mathcal{A} \to \mathcal{A}$, $a\not = b$, be the letter-to-letter map defined by $\gamma_{ab} (a) = \gamma_{ab} (b) = a$ and $\gamma_{ab} (c) =c$ for $c\in \mathcal{A}\setminus \{ a,b\}$.
Let $(a_n)_{1\leq n\leq N}$ be such that $\{ a_n \mid 1\leq n \leq N \} = \mathcal{A}$.
Then, $\gamma_{a_1 a_{2}} \gamma_{a_2 a_{3}} \cdots \gamma_{a_{N-1} a_{N}}$ is constant.
\end{lemma}
\begin{proof}
It suffices to observe that $\gamma_{a_m a_{m+1}} \gamma_{a_{m+1} a_{m+2}} \cdots \gamma_{a_{n-1} a_{n}}$ identifies the letters $a_m, a_{m+1}, a_{m+2} \dots , a_n$ to $a_m$.
\end{proof}
\begin{lemma}
Brun subshifts are primitive unimodular proper $\mathcal{S}$-adic subshifts.
\end{lemma}
\begin{proof}
Let $(X,S)$ be a Brun subshift over the alphabet $\mathcal{A}$, generated by the directive sequence $\boldsymbol{\beta } = (\beta_{a_nb_n})_{n\geq 1} \in S_\mathrm{Br}^\mathbb{N}$.
With each endomorphism $\beta_{ab}$ one can associate the map $\gamma : \mathcal{A} \to \mathcal{A}$ defined by $\gamma (c) = \beta (c)_0$.
Clearly $\gamma $ is equal to $\gamma_{ab}$.
By primitiveness, there exists an increasing sequence of integers $(n_k)_k$, with $n_0 =0$, such that $\{ a_i \mid n_k \leq i < n_{k+1} \} = \mathcal{A}$ for all $k$.
Hence from Lemma \ref{lemma:brunproper} the morphisms $\boldsymbol{\beta}_{[n_k , n_{k+1})}$ are left proper.
We conclude by using Lemma~\ref{lemma:proper}.
\end{proof}
As a corollary, we recover the following result (which also follows from~\cite[Theorem~5.7]{Berthe&Delecroix:14}).
\begin{proposition}\label{prop:UE}
Brun subshifts are uniquely ergodic.
\end{proposition}
\begin{proof}
This follows from Corollary~\ref{coro:measures} and from the fact that Brun subshifts have a simplex of letter measures generated by a single vector (see~\cite[Theorem~3.5]{Brentjes:81}).
\end{proof}
Brun subshifts have been introduced in \cite{BST:2019} in order to provide symbolic models for two-dimensional toral translations.
In particular, they are proved to have generically pure discrete spectrum in \cite{BST:2019}.
\subsection{Arnoux-Rauzy subshifts} \label{subsec:AR}
A minimal subshift $(X,S)$ over $ \mathcal{A}= \{1,2,\ldots,d\}$ is an \emph{Arnoux-Rauzy subshift} if for all~$n$ it has $(d-1)n+1$ factors of length~$n$, with exactly one left special and one right special factor of length~$n$.
Consider the following set of endomorphisms defined on the alphabet $\mathcal{A} = \{1, \dots , d \}$, namely
$S_\mathrm{AR} = \{ \alpha_a \mid a \in \mathcal{A}\}$ with
\[
\alpha_a:\ a \mapsto a,\ b \mapsto ab\ \mbox{for}\ b \in \mathcal{A} \setminus \{a\}.
\]
A subshift $(X,S)$ generated by a primitive directive sequence $\boldsymbol{\tau } \in S_\mathrm{AR}^\mathbb{N}$ is called an \emph{Arnoux-Rauzy subshift}.
It is standard to check that primitiveness of $\boldsymbol{\alpha }$ is equivalent to the fact that each morphism $\alpha_a$ occurs infinitely often in $\boldsymbol{\alpha }$.
Arnoux-Rauzy subshifts being dendric subshifts, they are in particular primitive unimodular proper $\mathcal{S}$-adic subshifts.
We similarly recover, as in Proposition~\ref{prop:UE}, that Arnoux-Rauzy subshifts are uniquely ergodic (see~\cite[Lemma 2]{Delecroix&Hejda&Steiner:2013} for
the fact that Arnoux-Rauzy subshifts have a simplex of letter measures generated by a single vector).
\subsection{A dendric subshift with non-trivial infinitesimals} \label{ex:nontrivial}
Let us provide an example of a minimal dendric subshift with non-trivial infinitesimal subgroup, and thus with rationally dependent letter measures according to
Proposition \ref{theo:saturated}.
We take the interval exchange $T$ with permutation $(1,3,2)$ with intervals $[0,1-2 \alpha) , [1-2 \alpha, 1- \alpha), $ and $[1-\alpha,1)$, with $\alpha=(3-\sqrt{5})/2$.
The transformation $T$ is represented in Figure~\ref{figure3interval}, with
$I_{1}=[0,1-2\alpha), $ $I_{2}=[1-2\alpha,1-\alpha)$, $ I_{3}=[1-\alpha,1)$
and $
J_{1}=[0,\alpha)$, $J_{2}=[\alpha,2\alpha)$, $J_{3}=[2\alpha,1).$
\begin{figure}
\caption{The transformation $T$.}
\label{figure3interval}
\end{figure}
Measures of letters are rationally dependent and the natural coding of this interval exchange is a strictly ergodic dendric subshift $(X,S,\mu)$ by Theorem \ref{theo:ergodic}.
It is actually a representation on $3$ intervals of the rotation of angle $2\alpha$
(the point $1-\alpha$ is a separation point which is not a singularity of this interval exchange).
One has
$\mu([2])=\mu([3])$.
The class of the function $\chi_{[2]}- \chi_{[3]}$ is thus a non-trivial infinitesimal, according to Theorem \ref{theo:cohoword}.
\subsection{Dendric subshifts having the same dimension group and different spectral properties } \label{ex:balance}
It is well known that within any given class of strong orbit equivalence (i.e., by Theorem \ref{oe}, within any family of minimal Cantor systems sharing the same dimension group $(G,G^+,u)$), all minimal Cantor systems share the same set of rational additive continuous eigenvalues $E(X,T)\cap \mathbb{Q}$ \cite{Ormes:97}.
When this set is reduced to $\{ 0 \}$, then, in the strong orbit equivalence class of $(X,T)$, there are many weakly mixing systems, see \cite[Theorem 6.1]{Ormes:97}, \cite[Theorem 5.4]{GHH:18} or \cite[Corollary 23]{Durand&Frank&Maass:2019}.
We provide here an example of a strong orbit equivalence class that contains two minimal dendric subshifts, one being weakly mixing and the other one having pure discrete spectrum.
Both systems are saturated (they have no non-trivial infinitesimals) but they have different balance properties.
They are defined on a three-letter alphabet and have factor complexity $2n + 1$.
According to Theorem \ref{theo:ergodic}, they are uniquely ergodic.
From Corollary~\ref{cor:oe}, two minimal dendric subshifts on a three-letter alphabet are strong orbit equivalent if and only if there is a unimodular row-stochastic matrix $M$ sending the vector of letter measures of one subshift to the vector of letter measures of the other.
In particular, any Arnoux-Rauzy subshift is strong orbit equivalent to any natural coding of an i.d.o.c. exchange of three intervals for which the length of the intervals are given by the letter measures of the Arnoux-Rauzy subshift (recall that an interval exchange transformation satisfies the {\it infinite distinct orbit condition}, i.d.o.c. for short, if the negative trajectories of the discontinuity points are infinite disjoint sets; this condition implies minimality \cite{Keane:75}). We thus consider the subshift $(X,S)$ generated by the Tribonacci substitution
$\sigma \colon a \mapsto ab, b \mapsto ac, c \mapsto a$ which is uniquely ergodic, dendric, balanced and has discrete spectrum \cite{Rauzy:1982}.
Let $\mu$ be its unique invariant measure.
We also consider the natural coding $(Y,S)$ of the three-letter interval exchange defined on intervals of length $\mu[a]$, $\mu[b]$, $\mu[c]$ with permutation $(13)(2)$.
It is uniquely ergodic, topologically weakly mixing \cite{KatokStepin,FerHol:2004} and strong orbit equivalent to $(X,S)$ by Proposition \ref{prop:DendricareSadic}.
Hence, for spectral reasons, $(X,S)$ and $(Y,S)$
are not topologically conjugate, even if they are strong orbit equivalent.
We provide a further proof of non-conjugacy for the systems $(X,S)$ and $(Y,S)$ based on asymptotic pairs.
We first recall a few definitions.
Two points $x,y $ in a given subshift are said to be {\em right asymptotic} if they have a common tail, i.e.,
there exists $n $ such that $(x_k)_{k \geq n}= (y_k)_{k \geq n}$.
This defines an equivalence relation on the collection of orbits:
two $S$-orbits ${\mathcal O}_S(x) = \{ S^n x \mid n\in \mathbb{Z} \} $ and ${\mathcal O}_S(y)$ are asymptotically equivalent
if for any $x' \in {\mathcal O}_S(x)$, there is $y' \in {\mathcal O}_S(y)$ that is right asymptotic to $x'$.
We call {\em asymptotic component} any equivalence class under the asymptotic equivalence.
We say that it is {\em non-trivial} whenever it is not reduced to one orbit.
An Arnoux-Rauzy subshift $(X,S)$ has a unique non-trivial asymptotic component formed of three distincts orbits as, for all $n$, there is a unique word $w$ of length $n$ such that $\ell(w) \geq 2$ and this word is such that $\ell(w)=3$ (see Section \ref{subsec:tree} for the notation).
On the other side, any i.d.o.c. exchange of three-intervals $(Y,S)$ has 2 asymptotic components and thus cannot be conjugate to $(X,S)$.
Indeed, suppose that it has a unique non-trivial asymptotic component.
As a natural coding of an i.d.o.c interval exchange transformation has two left special factors for each large enough length,
this component should contain three sequences $x'x$, $ x''ux$ and $x''' ux$ belonging to $Y$ where $u$ is a non-empty word.
This would imply that the interval exchange transformation is not i.d.o.c.
Next statement illustrates the variety of spectral behaviours within strong orbit equivalence classes of dendric subshifts.
\begin{proposition}\label{prop:realization}
For Lebesgue a.e. probability vector $\boldsymbol{ \mu}$ in ${\mathbb R}^3_+$,
there exist two strictly ergodic proper unimodular $\mathcal{S}$-adic subshifts, one with pure discrete spectrum and another one which is weakly mixing, both having the same dimension group
$\left( {\mathbb Z} ^3, \, \{ {\bf x} \in {\mathbb Z}^3 \mid \langle {\bf x}, \boldsymbol{ \mu} \rangle > 0 \}\cup \{{\mathbf 0}\},\, \bf{1}\right ).$
\end{proposition}
\begin{proof}
Brun subshifts such as introduced in Section \ref{subsec:Brun} are proved to have generically pure discrete spectrum
in
\cite{BST:2019}.
See \cite{KatokStepin,FerHol:2004} for the genericity of weak mixing for subshifts generated by three-letter interval exchanges.
\end{proof}
| 3,833 | 37,977 |
en
|
train
|
0.39.13
|
\subsection{Dendric vs. primitive unimodular proper $\mathcal{S}$-adic subshifts }\label{subsection:vs}
In this section, we give an example of a primitive unimodular proper $\mathcal{S}$-adic subshift whose strong orbit equivalence class contains no minimal dendric subshift.
Theorem~\ref{theo:dg} provides a description of the dimension group of any primitive unimodular proper $\mathcal{S}$-adic subshift.
It is natural to ask whether a strong orbit equivalence class represented by such a dimension group includes a primitive unimodular proper $\mathcal{S}$-adic subshift.
This was conjectured in different terms in \cite{Effros&Shen:1979}.
It was shown to be true when the dimension group has a unique trace~\cite{Riedel:1981b} (or, equivalently, when all minimal systems in this class are uniquely ergodic) but shown to be false in general~\cite{Riedel:1981}. In the same spirit, one may ask if the strong orbit equivalence class of any primitive unimodular proper $\mathcal{S}$-adic subshift contains a dendric subshift.
Inspired from \cite{Effros&Shen:1979}, we negatively answer to that question below.
Indeed, this example provides a family of examples of primitive unimodular $\mathcal{S}$-adic subshifts on a three-letter alphabet with two ergodic invariant probability measures.
They thus cannot be dendric by Theorem~\ref{theo:ergodic} and their strong orbit equivalence class contains no minimal dendric subshift.
Let ${\mathcal A}=\{1,2,3\}$ and consider the directive sequence $\boldsymbol{\tau } = (\tau_n : {\mathcal A}^* \to {\mathcal A}^*)_{n \geq 1}$ defined by
\begin{align*}
\tau_{2n}:
&
\
1 \mapsto 2^{a_n}3, \quad
2 \mapsto 1, \quad
3 \mapsto 2 \\
\tau_{2n+1}:
&
\
1 \mapsto 32^{a_n}, \quad
2 \mapsto 1, \quad
3 \mapsto 2
\end{align*}
where $(a_n)_{n \geq 1}$ is an increasing sequence of positive integers satisfying $\sum_{n \geq 1} 1/a_n <1$.
The incidence matrix of each morphism $\tau_n$ is the unimodular matrix
$$
A_n =
\begin{pmatrix}
0 & 1 & 0 \\
a_n & 0 & 1 \\
1 & 0 & 0 \\
\end{pmatrix}.
$$
It is easily checked that any morphism $\tau_{[n,n+5)}$ is proper and has an incidence matrix with positive entries.
Therefore, the $\mathcal{S}$-adic subshift $(X_{\boldsymbol{\tau}} , S)$ is primitive, unimodular and proper.
Let us show that $(X_{\boldsymbol{\tau}} , S)$ has two ergodic measures. In fact, we prove that $(X_{\boldsymbol{\tau}} , S)$ has at least two ergodic measures.
This will imply that it has exactly two ergodic measures by Theorem \ref{prop:effrosshen81}.
For all $n \geq 1$, let $C_n$ be the first column vector of $A_{[1,n]} = A_1 \cdots A_n$ and $J_n = C_n/\Vert C_n\Vert_1$ where $\Vert\cdot\Vert_1$ stands for the L1-norm.
If $(X_{\boldsymbol{\tau}} , S)$ had only one ergodic measure $\mu$, then $(J_n)_{n \geq 1}$ would converge to $\boldsymbol{\mu}$.
Hence it suffices to show that $(J_n)_{n \geq 1}$ does not converge.
Observe that, using the shape of the matrices $A_n$, that for $n\geq 1$, and setting $C_0 = e_1, C_{-1} = e_2, C_{-2} = e_3$, one has
$$
C_n = a_n C_{n-2} + C_{n-3},
$$
where $e_1,e_2,e_3$ are the canonical vectors.
Hence we have $J_n = b_n J_{n-2} + c_nJ_{n-3}$ with
$b_n = a_n \frac{\Vert C_{n-2}\Vert_1}{\Vert C_{n}\Vert_1}$ and
$c_n = \frac{\Vert C_{n-3}\Vert_1}{\Vert C_{n}\Vert_1}$.
In particular, $b_n + c_n = 1$.
As $(\Vert C_n\Vert_1)_n$ is non-decreasing, we have
$1 \geq b_n \geq a_n c_n$
and thus $c_n \leq a_n^{-1}$.
Moreover, we have
\[
\Vert J_n - J_{n-2} \Vert_1 = \Vert(b_n - 1) J_{n-2} -c_n J_{n-3} \Vert_1 = c_n \Vert J_{n-2} - J_{n-3} \Vert_1 \leq \frac{2}{a_n},
\]
hence, for $0\leq m\leq n$,
$$
\Vert J_{2n}-J_{2m}\Vert_1 \leq 2 \sum_{k=m+1}^{n} \frac{1}{a_{2k}}.
$$
This shows that $(J_{2n})_{n \geq 1}$ is a Cauchy sequence. Let $\beta$ stand for its limit.
For $n \geq m=0$, we obtain
$\Vert J_{2n}-e_1\Vert_1 \leq 2 \sum_{k=1}^{n} \frac{1}{a_{2k}}$ and thus
$\Vert\beta-e_1\Vert_1 \leq 2 \sum_{k=1}^{\infty } \frac{1}{a_{2k}}$.
We similarly show that $(J_{2n+1})_{n \geq 1}$ is a Cauchy sequence. Let $\alpha $ stand for its limit. We have $\Vert\alpha-e_2\Vert_1 \leq 2 \sum_{k=0}^{\infty } \frac{1}{a_{2k+1}}$.
Consequently,
$$
\Vert\alpha - \beta \Vert_1 =
\Vert(\alpha -e_2) + (e_1- \beta ) +e_2 -e_1 \Vert_1 \geq 2 - 2 \sum_{k=1}^{\infty } \frac{1}{a_{k}}
$$
and $(J_n)_{n \geq 1}$ does not converge.
Consequently, $(X_{\boldsymbol{\tau}} , S)$ has exactly two ergodic measures.
| 1,716 | 37,977 |
en
|
train
|
0.39.14
|
\section{Questions and further works}
According to \cite{Riedel:1981} (see also Section \ref{subsection:vs}), not all strong orbit equivalence classes represented by dimension groups of the type \eqref{align:ZdGD} in Theorem~\ref{theo:dg} contain primitive unimodular proper $\mathcal{S}$-adic subshifts.
The description of the dynamical dimension group in Theorem~\ref{theo:dg} is not precise enough to explain the restrictions that occur for instance for the measures,
so that a complete characterization of the dynamical dimension groups of primitive unimodular proper $\mathcal{S}$-adic subshifts is still missing.
Similarly, we address the question of characterizing the strong orbit equivalence classes containing minimal dendric subshifts.
The combinatorial properties of these subshifts imply constraints, especially for the invariant measures, such as stated in Theorem \ref{theo:ergodic}.
For example, the question arises as to whether dimension groups of rank $d$ having at most $d/2$ extremal traces are dimension groups of minimal dendric subshifts.
Another question is about the properness assumption.
For dendric or Brun subshifts, we were able to find a primitive unimodular proper $\mathcal{S}$-adic representation.
One can easily define $\mathcal{S}$-adic subshifts by a primitive unimodular directive sequence that is not proper.
The question now is whether
a primitive unimodular proper $\mathcal{S}$-adic representation (up to conjugacy) of this subshift can be found.
Even in the substitutive case, we do not know whether such a representation exists.
The factor complexity of dendric subshifts is affine.
It is well known~\cite{Boyle&Handelman:94,Ormes:97,Sugisaki:2003} that inside the strong orbit equivalence class of any minimal Cantor system one can find another minimal Cantor systems with any other prescribed topological entropy.
Primitive unimodular proper $\mathcal{S}$-adic subshifts being of finite topological rank, they have zero topological entropy.
It would be interesting to exhibit a variety of asymptotic behaviours for complexity functions within a strong orbit equivalence class.
\end{document}
| 572 | 37,977 |
en
|
train
|
0.40.0
|
\begin{equation}gin{document}
\title{Recurrence of biased quantum walks on a line}
\author{M. \v Stefa\v n\'ak$^{(1)}$, T. Kiss$^{(2)}$ and I. Jex$^{(1)}$}
\address{$^{(1)}$ Department of Physics, FJFI \v CVUT v Praze, B\v
rehov\'a 7, 115 19 Praha 1 - Star\'e M\v{e}sto, Czech Republic}
\address{$^{(2)}$ Department of Nonlinear and Quantum Optics, Research
Institute for Solid State Physics and Optics, Hungarian Academy of
Sciences, Konkoly-Thege u.29-33, H-1121 Budapest, Hungary}
\pacs{03.67.-a,05.40.Fb,02.30.Mv}
\date{\today}
\begin{equation}gin{abstract}
The P\'olya number of a classical random walk on a regular lattice is known to
depend solely on the dimension of the lattice. For one and two dimensions it equals one, meaning unit probability to return to the origin. This result is
extremely sensitive to the directional symmetry, any deviation
from the equal probability to travel in each direction results in a
change of the character of the walk from recurrent to transient.
Applying our definition of the P\'olya number to quantum walks on a
line we show that the recurrence character of quantum walks is more stable against bias. We determine the range of parameters for which biased quantum walks remain recurrent. We find that there exist genuine biased
quantum walks which are recurrent.
\end{abstract}
\maketitle
\section{Introduction}
\label{sec1}
Random walks are a popular topic in physics \cite{overview,hughes}. The popularity
stems from several sources. First, random walks are rather simple in their
formulation yet powerful in their application and allow to pinpoint the essential
physics involved in the studied processes. Next, the random walks are
one of the tools which allow to connect the microdynamics with the
macrobehaviour of large systems. Finally, random walks are quite
flexible and popular also outside physics to describe various phenomena. Hence it comes not as a surprise
that the first random walks have not been formulated within physics but to describe the alternation of share prices on the stock
exchange or the spreading of insects in a forest \cite{bachelier,chandrasekhar:1943}.
The study of random walks obtained a new stimulus when they were combined with quantum mechanics \cite{aharonov,konno:book}. Here the walker is thought to be a non-classical object enriched with wave attributes. The novel features of
quantum walks have been shown to be not only of theoretical interest
but to also have practical implications, especially for quantum algorithms \cite{kempe,shenvi:2003,aurel:2007,santha}. An important concept is the hitting time \cite{kempe:2005,krovi:2006a,krovi:2006b,magniez} which helps to point out the fundamental difference between classical and quantum walks allowing for algorithmic speed-up. One of the simplest non-trivial examples for a quantum walk is the one on a line \cite{nayak} which is closely related to the so called optical Galton board \cite{optical:galton}. Various aspects of one dimensional quantum walks have been analyzed \cite{tregenna,wojcik,knight,carteret,chandrashekar:2007,konno:2002}. Additional interesting effects, e.g. localisation, arise when one considers multi-state quantum walks \cite{1dloc1,1dloc,miyazaki,sato}.
One of the characteristics of the random walk on an infinite lattice is
expressed by the probability of the walker to return to its starting position, called the P\'olya number \cite{polya}. If the P\'olya number equals one
the walk is called recurrent, otherwise there is a non-zero probability that the walker never returns to its starting position. Such walks are called transient. The recurrence nature of the random walk is determined by the asymptotic behaviour of the probability at the origin \cite{revesz}. One finds that a random walk is transient if the probability at the origin decays faster than $t^{-1}$. The recurrent behaviour has been studied in great detail for classical random walks in dependence on the dimension and the topology of the lattice \cite{domb:1954,montroll:1964}.
Recently, we have extended the concept of P\'olya number to quantum walks \cite{prl}. In our definition we proposed a particular measurement scheme to minimize disturbance: each measurement in a series is carried out on a different member of an ensemble of equally prepared quantum systems. We have shown that the recurrence nature of the quantum walk, according to the above definition, is determined by the asymptotic behaviour of the probability at the origin in a similar way as in classical random walks. However, due to interference the asymptotics of the probability at the origin does not depend solely on the dimension of the lattice, but also on the coin operator and the initial coin state. Hence, one can find strikingly different recurrence behaviour for quantum walks compared to their classical counterpart \cite{pra}. Note that recurrence is meant here as the return to the origin which can be considered as a fractional recurrence from the point of view of the whole quantum state \cite{peres,chandra:recurrence} So far we have considered balanced walks, i.e. ones where there is no preference in direction for the walker and the step lengths are equal. For a large class of quantum walks this assumption does not hold and we wish to study the implications of unbalanced coins and unequal step lengths for the recurrence properties.
In the present paper we study biased quantum walks on the line
and compare their properties with their classical counterparts. As we briefly review in \ref{app:a}, recurrence of classical random walks is a consequence of the walk's symmetry. They are recurrent if and only if the mean value
of the position of the particle vanishes. This is due to the fact
that the spreading of the probability distribution of the
position is diffusive while the mean
value of the position propagates with a constant velocity. In contrast, for
quantum walks both the spreading of the probability distribution and
the propagation of the mean value are ballistic. We show that this allows for maintaining recurrence even when the symmetry is broken.
Our paper is organized as follows: In Section~\ref{sec2} we describe
the biased quantum walk on a line. In Section~\ref{sec3} we solve
the time evolution equations with the help of the Fourier
transformation. We find that the probability amplitudes can be
expressed in terms of integrals where time enters only in the
rapidly oscillating phase factor. This fact allows a straightforward
asymptotic analysis of the probability amplitudes by means of the
method of stationary phase. We perform this analysis in
Section~\ref{sec4}. Since the recurrence of the quantum walk is
determined by the asymptotics of the probability at the origin we
find a condition under which the biased quantum walk on a line is
recurrent. In Section~\ref{sec5} we analyze the recurrence of biased
quantum walks from a different perspective. We find that the
recurrence is related to the velocities of the peaks of the
probability distribution generated by the quantum walk. The explicit
form of the velocities leads us to the same condition derived in
Section~\ref{sec4}. Finally, in Section~\ref{sec6} we analyze the
formula for the mean value of the position of the particle derived
in \ref{app:b} in dependence of the parameters of the walk
and the initial state. We find that there exist genuine biased
quantum walks which are recurrent. Conclusions and outlook are
left for Section~\ref{sec7}.
\section{Description of the walk}
\label{sec2}
Let us consider biased quantum walks on a line where the particle has two possibilities --- jump to the right or to the left. Without loss of generality we restrict ourselves to biased quantum walks where the jump to the right is of the length $r$ and the jump to
the left has a unit size. We depict the biased quantum walk schematically in \fig{fig1}.
\begin{equation}gin{figure}
\begin{equation}gin{center}
\includegraphics[width=0.6\textwidth]{fig1.eps}
\caption{Schematics of the biased quantum walk on a line. If the
coin is in the state $|R\rangle$ the particle moves to the right
to a point at distance $r$. With the coin state $|L\rangle$
the particle makes a unit length step to the left. Before the step
itself the coin state is rotated according to the coin operator
$C(\rho)$.}
\label{fig1}
\end{center}
\end{figure}
The Hilbert space of the particle has the form of the tensor product
\begin{equation}gin{equation}
{\cal H} = {\cal H}_P\otimes{\cal H}_C
\end{equation}
of the position space
\begin{equation}gin{equation}
{\cal H}_P = \ell^2(\mathds{Z}) = \textrm{Span}\left\{|m\rangle|\ m\in\mathds{Z}\right\},
\end{equation}
and the two dimensional coin space
\begin{equation}gin{equation}
{\cal H}_C = \mathds{C}^2 = \textrm{Span}\left\{|R\rangle,|L\rangle\right\}.
\end{equation}
A single step of the quantum walk is given by the propagator
\begin{equation}gin{equation}
U = S \left(I_P\otimes C\right).
\label{qw:time}
\end{equation}
Here $I_P$ denotes the unit operator acting on the position space $\mathcal{H}_P$. The displacement operator $S$ has the form
\begin{equation}gin{equation}
S = \sum\limits_{m=-\infty}^{+\infty}|m+r\rangle\langle m|\otimes|R\rangle\langle R|+\sum\limits_{m=-\infty}^{+\infty}|m-1\rangle\langle m|\otimes|L\rangle\langle L|.
\end{equation}
The coin flip $C$ is in general an arbitrary unitary operator acting on the coin space $\mathcal{H}_C$ and is applied on the
coin state before the displacement $S$ itself. However, as has been discussed in \cite{tregenna} the probability distribution is not affected by the complex phases of the coin operator. Hence, it is sufficient to consider the one-parameter
family of coins
\begin{equation}gin{equation}
C(\rho) = \left(
\begin{equation}gin{array}{cc}
\sqrt{\rho} & \sqrt{1-\rho} \\
\sqrt{1-\rho} & -\sqrt{\rho} \\
\end{array}
\right).
\label{coins}
\end{equation}
From now on we restrict ourselves to this family
of coins. The value of $\rho=1/2$ corresponds to the well known case of the Hadamard walk.
We write the initial state of the particle in the form
\begin{equation}gin{equation}
|\psi(0)\rangle \equiv
\sum\limits_{m=-\infty}^{+\infty}\sum_{i=R}^L\psi_i(m,0)|m\rangle\otimes|i\rangle.
\end{equation}
The state of the walker after $t$ steps is given by successive
application of the time evolution operator given by Eq.
(\ref{qw:time}) on the initial state
\begin{equation}gin{equation}
|\psi(t)\rangle \equiv
U^t|\psi(0)\rangle =
\sum\limits_{m=-\infty}^{+\infty}\sum_{i=R}^L\psi_i(m,t)|m\rangle\otimes|i\rangle.
\label{time:evol}
\end{equation}
The state of the particle is fully
determined by the set of two-component vectors
\begin{equation}gin{equation}
\psi(m,t)\equiv{\left(\psi_R(m,t),\psi_L(m,t)\right)}^T.
\label{prob:ampl}
\end{equation}
Here $\psi_{R(L)}(m,t)$ is the probability
amplitude to find the particle at position $m$ after $t$ steps with
the coin state $|R(L)\rangle$. The probability distribution
generated by the quantum walk is given by
\begin{equation}gin{eqnarray}
\nonumber P(m,t) & = & |\langle m,R|\psi(t)\rangle|^2+|\langle m,L|\psi(t)\rangle|^2 \\
\nonumber & = & |\psi_R(m,t)|^2+|\psi_L(m,t)|^2 = \|\psi(m,t)\|^2.\\
\end{eqnarray}
| 3,109 | 16,635 |
en
|
train
|
0.40.1
|
\section{Description of the walk}
\label{sec2}
Let us consider biased quantum walks on a line where the particle has two possibilities --- jump to the right or to the left. Without loss of generality we restrict ourselves to biased quantum walks where the jump to the right is of the length $r$ and the jump to
the left has a unit size. We depict the biased quantum walk schematically in \fig{fig1}.
\begin{equation}gin{figure}
\begin{equation}gin{center}
\includegraphics[width=0.6\textwidth]{fig1.eps}
\caption{Schematics of the biased quantum walk on a line. If the
coin is in the state $|R\rangle$ the particle moves to the right
to a point at distance $r$. With the coin state $|L\rangle$
the particle makes a unit length step to the left. Before the step
itself the coin state is rotated according to the coin operator
$C(\rho)$.}
\label{fig1}
\end{center}
\end{figure}
The Hilbert space of the particle has the form of the tensor product
\begin{equation}gin{equation}
{\cal H} = {\cal H}_P\otimes{\cal H}_C
\end{equation}
of the position space
\begin{equation}gin{equation}
{\cal H}_P = \ell^2(\mathds{Z}) = \textrm{Span}\left\{|m\rangle|\ m\in\mathds{Z}\right\},
\end{equation}
and the two dimensional coin space
\begin{equation}gin{equation}
{\cal H}_C = \mathds{C}^2 = \textrm{Span}\left\{|R\rangle,|L\rangle\right\}.
\end{equation}
A single step of the quantum walk is given by the propagator
\begin{equation}gin{equation}
U = S \left(I_P\otimes C\right).
\label{qw:time}
\end{equation}
Here $I_P$ denotes the unit operator acting on the position space $\mathcal{H}_P$. The displacement operator $S$ has the form
\begin{equation}gin{equation}
S = \sum\limits_{m=-\infty}^{+\infty}|m+r\rangle\langle m|\otimes|R\rangle\langle R|+\sum\limits_{m=-\infty}^{+\infty}|m-1\rangle\langle m|\otimes|L\rangle\langle L|.
\end{equation}
The coin flip $C$ is in general an arbitrary unitary operator acting on the coin space $\mathcal{H}_C$ and is applied on the
coin state before the displacement $S$ itself. However, as has been discussed in \cite{tregenna} the probability distribution is not affected by the complex phases of the coin operator. Hence, it is sufficient to consider the one-parameter
family of coins
\begin{equation}gin{equation}
C(\rho) = \left(
\begin{equation}gin{array}{cc}
\sqrt{\rho} & \sqrt{1-\rho} \\
\sqrt{1-\rho} & -\sqrt{\rho} \\
\end{array}
\right).
\label{coins}
\end{equation}
From now on we restrict ourselves to this family
of coins. The value of $\rho=1/2$ corresponds to the well known case of the Hadamard walk.
We write the initial state of the particle in the form
\begin{equation}gin{equation}
|\psi(0)\rangle \equiv
\sum\limits_{m=-\infty}^{+\infty}\sum_{i=R}^L\psi_i(m,0)|m\rangle\otimes|i\rangle.
\end{equation}
The state of the walker after $t$ steps is given by successive
application of the time evolution operator given by Eq.
(\ref{qw:time}) on the initial state
\begin{equation}gin{equation}
|\psi(t)\rangle \equiv
U^t|\psi(0)\rangle =
\sum\limits_{m=-\infty}^{+\infty}\sum_{i=R}^L\psi_i(m,t)|m\rangle\otimes|i\rangle.
\label{time:evol}
\end{equation}
The state of the particle is fully
determined by the set of two-component vectors
\begin{equation}gin{equation}
\psi(m,t)\equiv{\left(\psi_R(m,t),\psi_L(m,t)\right)}^T.
\label{prob:ampl}
\end{equation}
Here $\psi_{R(L)}(m,t)$ is the probability
amplitude to find the particle at position $m$ after $t$ steps with
the coin state $|R(L)\rangle$. The probability distribution
generated by the quantum walk is given by
\begin{equation}gin{eqnarray}
\nonumber P(m,t) & = & |\langle m,R|\psi(t)\rangle|^2+|\langle m,L|\psi(t)\rangle|^2 \\
\nonumber & = & |\psi_R(m,t)|^2+|\psi_L(m,t)|^2 = \|\psi(m,t)\|^2.\\
\end{eqnarray}
\section{Time evolution of the walk}
\label{sec3}
To obtain explicit and closed form expressions for the time
dependent state vector we rewrite the time evolution equation
(\ref{time:evol}) for the state vector $|\psi(t)\rangle$ into a set
of difference equations
\begin{equation}gin{eqnarray}
\nonumber \psi(m,t) & = & C_+(\rho)\psi(m-r,t-1)+\\
& & +C_-(\rho)\psi(m+1,t-1)
\label{time:evol2}
\end{eqnarray}
for the probability amplitude vectors $\psi(m,t)$. The form of the
matrices $C_\pm(\rho)$ follows from the matrix $C(\rho)$
\begin{equation}gin{equation}
C_+(\rho) = \left(
\begin{equation}gin{array}{cc}
\sqrt{\rho} & \sqrt{1-\rho} \\
0 & 0 \\
\end{array}
\right),\qquad
C_-(\rho) = \left(
\begin{equation}gin{array}{cc}
0 & 0 \\
\sqrt{1-\rho} & -\sqrt{\rho} \\
\end{array}
\right).
\end{equation}
The time evolution equations (\ref{time:evol2}) are greatly
simplified with the help of the Fourier transformation
\begin{equation}gin{equation}
\tilde{\psi}(k,t)\equiv\sum\limits_{m=-\infty}^{+\infty}\psi(m,t)e^{i mk},
\label{qw:ft}
\end{equation}
where the momentum $k$ is a continuous parameter ranging from $-\pi$ to $\pi$. The new function
$\tilde{\psi}(k,t)$ is square integrable on a unit circle.
The time evolution in the Fourier picture turns into a single difference equation
\begin{equation}gin{equation}
\tilde{\psi}(k,t) = \widetilde{U}(k)\tilde{\psi}(k,t-1),
\label{qw:te:fourier}
\end{equation}
where the propagator has the form
\begin{equation}gin{equation}
\widetilde{U}(k) \equiv \left(
\begin{equation}gin{array}{cc}
\sqrt{\rho}e^{ikr} & \sqrt{1-\rho}e^{ikr} \\
\sqrt{1-\rho}e^{-ik} & -\sqrt{\rho}e^{-ik} \\
\end{array}
\right).
\label{teopF}
\end{equation}
The solution of (\ref{qw:te:fourier}) is
straightforward. We find
\begin{equation}gin{equation}
\tilde{\psi}(k,t) =\widetilde{U}^t(k)\tilde{\psi}(k,0),
\end{equation}
where $\tilde{\psi}(k,0)$ is the Fourier transformation of the initial
state. We restrict ourselves to the situation where the particle is
initially localized at the origin as dictated by the nature of
the problem we wish to study. As follows from (\ref{qw:ft}) the
Fourier transformation $\tilde{\psi}(k,0)$ of such an initial
condition is equal to the initial state of the coin
\begin{equation}gin{equation}
\tilde{\psi}(k,0) = \left(
\begin{equation}gin{array}{c}
\psi_R(0,0) \\
\psi_L(0,0) \\
\end{array}
\right),
\end{equation}
which we denote by $\psi$. Since $\psi$ can be an arbitrary
normalized complex two-component vector we parameterize it
by two parameters $a\in[0,1]$ and $\varphi\in[0,2\pi)$ in the
form
\begin{equation}gin{equation}
\psi = \left(
\begin{equation}gin{array}{c}
\sqrt{a} \\
\sqrt{1-a}e^{i\varphi} \\
\end{array}
\right).
\label{psi:init}
\end{equation}
To evaluate the powers of the propagator $\widetilde{U}(k)$
it is convenient to diagonalize it. Since the propagator is
unitary its eigenvalues have the form $e^{i\omega_{1,2}}$ where the
phases read
\begin{equation}gin{eqnarray}
\nonumber \omega_1(k) & = & \frac{r-1}{2}k+\arcsin\left(\sqrt{\rho}\sin\left(\frac{r+1}{2}k\right)\right),\\
\omega_2(k) & = & \frac{r-1}{2}k -\pi-\arcsin\left(\sqrt{\rho}\sin\left(\frac{r+1}{2}k\right)\right)
\label{omega}
\end{eqnarray}
We denote the corresponding eigenvectors by $v_{1,2}(k)$. We
give their explicit form in the \ref{app:b}. With this notation we write
the solution of the time evolution equation in the Fourier picture
in the form
\begin{equation}gin{equation}
\widetilde{\psi}(k,t) = \sum_{j=1}^2
e^{i\omega_j(k)t}\left(v_j(k),\psi\right)v_j(k).
\end{equation}
Here $( , )$ means scalar product in the coin space. Finally, we obtain the solution in position representation by
performing the inverse Fourier transformation
\begin{equation}gin{eqnarray}
\nonumber \psi(m,t) = \int_{-\pi}^\pi\frac{dk}{2\pi}\ \widetilde{\psi}(k,t)\ e^{-imk} = \sum_{j=1}^2\int_{-\pi}^\pi\frac{dk}{2\pi}e^{i(\omega_j(k)t-mk)}\ \left(v_j(k),\psi\right)v_j(k).\\
\label{inv:f}
\end{eqnarray}
\section{Asymptotics of the quantum walk and recurrence}
\label{sec4}
To determine the recurrence nature of the biased quantum walk we
have to analyze the asymptotic behaviour of the probability at the
origin \cite{prl}. Exploiting (\ref{inv:f}) the amplitude at the origin
reads
\begin{equation}gin{equation}
\psi(0,t) = \sum_{j=1}^2\int_{-\pi}^\pi\frac{dk}{2\pi}e^{i\omega_j(k)t}\
\left(v_j(k),\psi\right)v_j(k),
\label{psi:0}
\end{equation}
which allows us to find the asymptotics of the probability at the origin
with the help of the method of stationary phase \cite{statphase}.
The important contributions to the integrals in (\ref{psi:0}) arise
from the stationary points of the phases (\ref{omega}). We find that
the derivatives of the phases $\omega_{1,2}(k)$ are
\begin{equation}gin{eqnarray}
\nonumber \omega_1'(k) & = & \frac{r-1}{2}+\frac{\sqrt{\rho}(r+1)\cos\left(k\frac{r+1}{2}\right)}{\sqrt{4+2\rho\left[\cos(k(r+1))-1\right]}},\\
\nonumber \omega_2'(k) & = & \frac{r-1}{2}-\frac{\sqrt{\rho}(r+1)\cos\left(k\frac{r+1}{2}\right)}{\sqrt{4+2\rho\left[\cos(k(r+1))-1\right]}}.\\
\label{phase:der}
\end{eqnarray}
Using the method of stationary phase we find that the
amplitude will decay slowly - like $t^{-\frac{1}{2}}$, if at least
one of the phases has a vanishing derivative inside the integration
domain. Solving the equations $\omega_{1,2}'(k) = 0$ we find
that the possible saddle points are
\begin{equation}gin{equation}
k_0 = \pm\frac{2}{r+1}\arccos\left(\pm\sqrt{\frac{(1-\rho)(r-1)^2}{4\rho r}}\right).
\label{k0}
\end{equation}
The saddle points are real valued
provided the argument of the arcus-cosine in (\ref{k0}) is less or
equal to unity
\begin{equation}gin{equation}
\frac{(1-\rho)(r-1)^2}{4\rho r} \leq 1.
\end{equation}
This inequality leads us to the condition for the biased quantum
walk on a line to be recurrent
\begin{equation}gin{equation}
\rho_R(r) \geq \left(\frac{r-1}{r+1}\right)^2.
\label{crit:rec}
\end{equation}
We illustrate this result in \fig{fig2} for a particular choice of the walk parameter $r=3$.
\begin{equation}gin{figure}[h]
\begin{equation}gin{center}
\includegraphics[width=0.5\textwidth]{fig2.eps}
\caption{The existence of stationary points of the phases
$\omega_{1,2}(k)$ in dependence on the parameter $\rho$ and
a fixed step length $r$. We plot the implicit functions $\omega_{1,2}'(k)\equiv
0$ for $r=3$. The plot indicates that for
$\rho<\rho_R(3)=\frac{1}{4}$ the phases $\omega_{1,2}(k)$ do
not have any saddle points. Consequently, the probability amplitude
at the origin decays fast and such biased quantum walk on a line is
transient. For $\rho\geq\rho_R(3)$ the saddle points exist and the
quantum walk is recurrent.}
\label{fig2}
\end{center}
\end{figure}
Our simple result proves that there is an intimate nontrivial
link between the length of the step of the walk and the bias of the
coin. The parameter of the coin $\rho$ has to be at least equal to
a factor determined by the size of the step to the right $r$ for the
walk to be recurrent. We note that the recurrence nature of the
biased quantum walk on a line is determined only by the parameters
of the walk itself, i.e. the coin and the step, not by the initial
conditions. The parameters of the initial state $a$ and $\varphi$
have no effect on the rate of decay of the probability at the
origin.
| 3,846 | 16,635 |
en
|
train
|
0.40.2
|
\section{Asymptotics of the quantum walk and recurrence}
\label{sec4}
To determine the recurrence nature of the biased quantum walk we
have to analyze the asymptotic behaviour of the probability at the
origin \cite{prl}. Exploiting (\ref{inv:f}) the amplitude at the origin
reads
\begin{equation}gin{equation}
\psi(0,t) = \sum_{j=1}^2\int_{-\pi}^\pi\frac{dk}{2\pi}e^{i\omega_j(k)t}\
\left(v_j(k),\psi\right)v_j(k),
\label{psi:0}
\end{equation}
which allows us to find the asymptotics of the probability at the origin
with the help of the method of stationary phase \cite{statphase}.
The important contributions to the integrals in (\ref{psi:0}) arise
from the stationary points of the phases (\ref{omega}). We find that
the derivatives of the phases $\omega_{1,2}(k)$ are
\begin{equation}gin{eqnarray}
\nonumber \omega_1'(k) & = & \frac{r-1}{2}+\frac{\sqrt{\rho}(r+1)\cos\left(k\frac{r+1}{2}\right)}{\sqrt{4+2\rho\left[\cos(k(r+1))-1\right]}},\\
\nonumber \omega_2'(k) & = & \frac{r-1}{2}-\frac{\sqrt{\rho}(r+1)\cos\left(k\frac{r+1}{2}\right)}{\sqrt{4+2\rho\left[\cos(k(r+1))-1\right]}}.\\
\label{phase:der}
\end{eqnarray}
Using the method of stationary phase we find that the
amplitude will decay slowly - like $t^{-\frac{1}{2}}$, if at least
one of the phases has a vanishing derivative inside the integration
domain. Solving the equations $\omega_{1,2}'(k) = 0$ we find
that the possible saddle points are
\begin{equation}gin{equation}
k_0 = \pm\frac{2}{r+1}\arccos\left(\pm\sqrt{\frac{(1-\rho)(r-1)^2}{4\rho r}}\right).
\label{k0}
\end{equation}
The saddle points are real valued
provided the argument of the arcus-cosine in (\ref{k0}) is less or
equal to unity
\begin{equation}gin{equation}
\frac{(1-\rho)(r-1)^2}{4\rho r} \leq 1.
\end{equation}
This inequality leads us to the condition for the biased quantum
walk on a line to be recurrent
\begin{equation}gin{equation}
\rho_R(r) \geq \left(\frac{r-1}{r+1}\right)^2.
\label{crit:rec}
\end{equation}
We illustrate this result in \fig{fig2} for a particular choice of the walk parameter $r=3$.
\begin{equation}gin{figure}[h]
\begin{equation}gin{center}
\includegraphics[width=0.5\textwidth]{fig2.eps}
\caption{The existence of stationary points of the phases
$\omega_{1,2}(k)$ in dependence on the parameter $\rho$ and
a fixed step length $r$. We plot the implicit functions $\omega_{1,2}'(k)\equiv
0$ for $r=3$. The plot indicates that for
$\rho<\rho_R(3)=\frac{1}{4}$ the phases $\omega_{1,2}(k)$ do
not have any saddle points. Consequently, the probability amplitude
at the origin decays fast and such biased quantum walk on a line is
transient. For $\rho\geq\rho_R(3)$ the saddle points exist and the
quantum walk is recurrent.}
\label{fig2}
\end{center}
\end{figure}
Our simple result proves that there is an intimate nontrivial
link between the length of the step of the walk and the bias of the
coin. The parameter of the coin $\rho$ has to be at least equal to
a factor determined by the size of the step to the right $r$ for the
walk to be recurrent. We note that the recurrence nature of the
biased quantum walk on a line is determined only by the parameters
of the walk itself, i.e. the coin and the step, not by the initial
conditions. The parameters of the initial state $a$ and $\varphi$
have no effect on the rate of decay of the probability at the
origin.
\section{Recurrence of a quantum walk and the velocities of the peaks}
\label{sec5}
We can determine the recurrence nature of the biased quantum walk on
a line from a different point of view. This approach is based
on the following observation. The well known shape of the
probability distribution generated by the quantum walk consists of
two counter-propagating peaks. In between
the two dominant peaks the probability is roughly independent of $m$ and decays like $t^{-1}$.
On the other hand, outside the decay is exponential as we depart from the peaks. As it has been found in \cite{nayak} the position of the peaks varies linearly with the number of steps. Hence, the peaks propagate with constant velocities, say $v_L$ and $v_R$. For the
biased quantum walk to be recurrent the origin of the walk has to
remain in between the two peaks for all times. In other words, the
biased quantum walk on a line is recurrent if and only if the
velocity of the left peak is negative and the velocity of the right
peak is positive.
The velocities of the left and right peaks are easily determined. We
rewrite the formula (\ref{inv:f}) for the probability amplitude
$\psi(m,t)$ into the form
\begin{equation}gin{equation}
\psi(m,t) = \sum_{j=1}^2\int_{-\pi}^\pi\frac{dk}{2\pi}e^{i(\omega_j(k)-\alpha
k)t}\ \left(v_j(k),\psi\right)v_j(k),
\end{equation}
where we have introduced $\alpha = \frac{m}{t}$. Due to the fact that we now
concentrate on the amplitudes at the positions $m\sim t$ we have to
consider modified phases
\begin{equation}gin{equation}
\widetilde{\omega}_j(k) = \omega_j(k)-\alpha k.
\end{equation}
The peak occurs at such a position
$m_0$ where both the first and the second derivatives of
$\widetilde{\omega}_j(k)$ vanishes. The velocity of the peak is
thus $\alpha_0 = \frac{m_0}{t}$. Hence, solving the equations
\begin{equation}gin{eqnarray}
\nonumber \widetilde{\omega}_1'(k) & = & \frac{r-1}{2}+\frac{\sqrt{\rho}(r+1)\cos\left(k\frac{r+1}{2}\right)}{\sqrt{4+2\rho\left[\cos(k(r+1))-1\right]}} - \alpha = 0 ,\\
\nonumber \widetilde{\omega}_2'(k) & = & \frac{r-1}{2}-\frac{\sqrt{\rho}(r+1)\cos\left(k\frac{r+1}{2}\right)}{\sqrt{4+2\rho\left[\cos(k(r+1))-1\right]}} - \alpha = 0,\\
\nonumber \widetilde{\omega}_1''(k) & = & -\widetilde{\omega}_2''(k) = \frac{(\rho-1)\sqrt{\rho}(r+1)^2\sin\left(k\frac{r+1}{2}\right)}{\sqrt{2}\left[2-\rho+\rho\cos(k(r+1))\right]^\frac{3}{2}} = 0,\\
\end{eqnarray}
for $\alpha$ determines the velocities of the left and right peak $v_{L,R}$. The third equation is independent of $\alpha$ and we easily find the
solution
\begin{equation}gin{equation}
k_0 = \frac{4n\pi}{r+1},\ k_0=\frac{2\pi(2n+1)}{r+1},\ n\in\mathds{Z}.
\end{equation}
Inserting this $k_0$ into the first two equations we find the velocities of the left and right peak
\begin{equation}gin{eqnarray}
\nonumber v_L & = & \frac{r-1}{2}-\frac{r+1}{2}\sqrt{\rho}\\
v_R & = & \frac{r-1}{2}+\frac{r+1}{2}\sqrt{\rho}.
\label{velocities}
\end{eqnarray}
We illustrate this result in \fig{fig3} where we show the probability distribution generated by the quantum walk for the particular choice of the parameters $r = 3,\ \rho = \frac{1}{\sqrt{2}}$. The initial state was chosen according to $a = \frac{1}{\sqrt{2}}$ and $\varphi = \pi$. Since the velocity of the left peak $v_L$ is negative this biased quantum walk is recurrent.
\begin{equation}gin{figure}[h]
\begin{equation}gin{center}
\includegraphics[width=0.7\textwidth]{fig3.eps}
\caption{Velocities of the left and right peak of the probability distribution generated by the biased quantum walk on a line and the recurrence.
We have chosen the parameters $r=3$, $a=\rho=\frac{1}{\sqrt{2}}$ and $\varphi = \pi$. The left peak propagates with the velocity $v_L\approx -0.68$,
the velocity of the right peak is $v_R\approx 2.68$. In between the two peaks the probability distribution behaves like $t^{-1}$ while outside the
decay is exponential. Since the velocity $v_L$ is negative the origin of the walk remains in between the left and right peak. Consequently, this quantum
walk is recurrent.}
\label{fig3}
\end{center}
\end{figure}
The peak velocities have two contributions. One is identical
and independent of $\rho$, the second is a product of $r$ and $\rho$
and differs in sign for the two velocities. The obtained results indicate that biasing the walk by having
the size of the step to the right equal to $r$ results in dragging
the whole probability distribution towards the direction of
the larger step. This is manifested by the term $\frac{r-1}{2}$
which appears in both velocities $v_{L,R}$ with the same sign. On
the other hand the parameter of the coin $\rho$ does not bias the
walk. As we can see from the second terms entering the velocities it
rather influences the rate at which the walk spreads.
As we have discussed above the biased quantum walk on a line is recurrent if and only if $v_L$ is negative and $v_R$ is positive. The form of the
velocities (\ref{velocities}) implies that this condition is satisfied if and only if the criterion (\ref{crit:rec}) is fulfilled.
\section{Mean value of the biased quantum walk and recurrence}
\label{sec6}
As we discuss in the \ref{app:a} the classical random
walks are recurrent if and only if the mean value of the position
vanishes. We now show that this is not true for biased quantum
walks, i.e. there exist biased quantum walks on a line which are
recurrent but cannot produce probability distribution with zero mean
value. This is another unique feature of quantum walks
compared to the classical ones.
In the \ref{app:b} we derive the following formula
for the position mean value
\begin{equation}gin{eqnarray}
\nonumber \left\langle\frac{x}{t}\right\rangle & \approx & (1-\sqrt{1-\rho})(a(r+1)-1)+\\
\nonumber & & +\frac{\sqrt{a(1-a)}(1-\sqrt{1-\rho})(1-\rho)(r+1)\cos\varphi}{\sqrt{\rho(1-\rho)}}\\
& & +\frac{r-1}{2}\sqrt{1-\rho}+O(t^{-1}).
\label{mean}
\end{eqnarray}
We see that for quantum walks the mean value is affected by both the
fundamental walk parameters through $r$ and $\rho$ and the
initial state parameters $a$ and $\varphi$. The mean value is
typically non-vanishing even for unbiased quantum walks (
with $r=1$ ). However, one easily finds \cite{tregenna} that the initial
state with the parameters $a=1/2$ and $\varphi=\pi/2$ results in a
symmetric probability distribution with zero mean independent of the
coin parameter $\rho$. Indeed, the quantum walks with $r=1$, i.e.
with equal steps to the right and left, do not intrinsically
distinguish left from right. On the other hand the quantum walks
with $r>1$ treat the left and right direction in a different way.
Nevertheless, one can always find for a given $r$ a coin parameter
$\rho_0$ such that for all $\rho\geq\rho_0$ the quantum walk can
produce a probability distribution with zero mean value. This is
impossible for quantum walks with $\rho<\rho_0$ and we will call
such quantum walks genuine biased.
Let us now determine the minimal value of $\rho$ for a given $r$ for
which mean value vanishes. We first find the parameters of the
initial state $a$ and $\varphi$ which minimizes the mean value.
Clearly the term on the second line in (\ref{mean}) reaches the
minimal value for $\varphi_0=\pi$. Differentiating the resulting
expression with respect to $a$ and setting the derivative equal to
zero gives us the condition
\begin{equation}gin{equation}
2 + \frac{(2a-1)
\sqrt{\rho(1-\rho)}}{\rho\sqrt{a(1-a)}} = 0
\end{equation}
on the minimal mean value with respect to $a$. This relation is satisfied for
$a_0=\frac{1}{2}(1-\sqrt{\rho})$. The resulting formula for the mean
value reads
\begin{equation}gin{equation}
\left\langle\frac{x}{t}\right\rangle_{a_0,\varphi_0}
= \frac{r-1}{2}+\frac{\left(1-\sqrt{1-\rho}-\rho\right) (1+r)}{2
\sqrt{(1-\rho) \rho}}.
\label{mean:min}
\end{equation}
This expression vanishes for
\begin{equation}gin{equation}
\rho_0(r) = \left(\frac{r^2 - 1}{r^2 + 1}\right)^2.
\label{rho:0}
\end{equation}
Since (\ref{mean:min}) is a decreasing function of
$\rho$ the mean value is always positive for $\rho<\rho_0$
independent of the choice of the initial state. For $\rho>\rho_0$
one can achieve zero mean value for different combination of the
parameters $a$ and $\varphi$.
The formula (\ref{rho:0}) is reminiscent of the condition
(\ref{crit:rec}) for the biased quantum walk on a line to be
recurrent. However, $r$ is in (\ref{rho:0}) replaced by $r^2$.
Therefore we find the inequality $\rho_R<\rho_0$. Hence, the quantum
walks with the coin parameter $\rho_R<\rho<\rho_0$ are recurrent but
cannot produce a probability distribution with zero mean value. We
conclude that there are genuine biased quantum walks which are
recurrent in contrast to situations found for classical walks.
| 3,802 | 16,635 |
en
|
train
|
0.40.3
|
\section{Mean value of the biased quantum walk and recurrence}
\label{sec6}
As we discuss in the \ref{app:a} the classical random
walks are recurrent if and only if the mean value of the position
vanishes. We now show that this is not true for biased quantum
walks, i.e. there exist biased quantum walks on a line which are
recurrent but cannot produce probability distribution with zero mean
value. This is another unique feature of quantum walks
compared to the classical ones.
In the \ref{app:b} we derive the following formula
for the position mean value
\begin{equation}gin{eqnarray}
\nonumber \left\langle\frac{x}{t}\right\rangle & \approx & (1-\sqrt{1-\rho})(a(r+1)-1)+\\
\nonumber & & +\frac{\sqrt{a(1-a)}(1-\sqrt{1-\rho})(1-\rho)(r+1)\cos\varphi}{\sqrt{\rho(1-\rho)}}\\
& & +\frac{r-1}{2}\sqrt{1-\rho}+O(t^{-1}).
\label{mean}
\end{eqnarray}
We see that for quantum walks the mean value is affected by both the
fundamental walk parameters through $r$ and $\rho$ and the
initial state parameters $a$ and $\varphi$. The mean value is
typically non-vanishing even for unbiased quantum walks (
with $r=1$ ). However, one easily finds \cite{tregenna} that the initial
state with the parameters $a=1/2$ and $\varphi=\pi/2$ results in a
symmetric probability distribution with zero mean independent of the
coin parameter $\rho$. Indeed, the quantum walks with $r=1$, i.e.
with equal steps to the right and left, do not intrinsically
distinguish left from right. On the other hand the quantum walks
with $r>1$ treat the left and right direction in a different way.
Nevertheless, one can always find for a given $r$ a coin parameter
$\rho_0$ such that for all $\rho\geq\rho_0$ the quantum walk can
produce a probability distribution with zero mean value. This is
impossible for quantum walks with $\rho<\rho_0$ and we will call
such quantum walks genuine biased.
Let us now determine the minimal value of $\rho$ for a given $r$ for
which mean value vanishes. We first find the parameters of the
initial state $a$ and $\varphi$ which minimizes the mean value.
Clearly the term on the second line in (\ref{mean}) reaches the
minimal value for $\varphi_0=\pi$. Differentiating the resulting
expression with respect to $a$ and setting the derivative equal to
zero gives us the condition
\begin{equation}gin{equation}
2 + \frac{(2a-1)
\sqrt{\rho(1-\rho)}}{\rho\sqrt{a(1-a)}} = 0
\end{equation}
on the minimal mean value with respect to $a$. This relation is satisfied for
$a_0=\frac{1}{2}(1-\sqrt{\rho})$. The resulting formula for the mean
value reads
\begin{equation}gin{equation}
\left\langle\frac{x}{t}\right\rangle_{a_0,\varphi_0}
= \frac{r-1}{2}+\frac{\left(1-\sqrt{1-\rho}-\rho\right) (1+r)}{2
\sqrt{(1-\rho) \rho}}.
\label{mean:min}
\end{equation}
This expression vanishes for
\begin{equation}gin{equation}
\rho_0(r) = \left(\frac{r^2 - 1}{r^2 + 1}\right)^2.
\label{rho:0}
\end{equation}
Since (\ref{mean:min}) is a decreasing function of
$\rho$ the mean value is always positive for $\rho<\rho_0$
independent of the choice of the initial state. For $\rho>\rho_0$
one can achieve zero mean value for different combination of the
parameters $a$ and $\varphi$.
The formula (\ref{rho:0}) is reminiscent of the condition
(\ref{crit:rec}) for the biased quantum walk on a line to be
recurrent. However, $r$ is in (\ref{rho:0}) replaced by $r^2$.
Therefore we find the inequality $\rho_R<\rho_0$. Hence, the quantum
walks with the coin parameter $\rho_R<\rho<\rho_0$ are recurrent but
cannot produce a probability distribution with zero mean value. We
conclude that there are genuine biased quantum walks which are
recurrent in contrast to situations found for classical walks.
\section{Conclusions}
\label{sec7}
We have analyzed one dimensional biased quantum walks. Classically, the bias leading to a non-zero mean value of the particle's position can be introduced in two ways --- unequal step lengths or unfair coin. In contrast, for quantum walks on a line the initial state can introduce bias for any coin. On the other hand, for symmetric initial state modifying only the unitary coin operator while keeping the equal step lengths will not introduce bias. Finally, the bias due to unequal step lengths may be compensated for by the choice of the coin operator for some initial conditions. For this reason we have introduced the concept of the genuinely biased quantum walk for which there does not exists any initial state leading to vanishing mean value of the position.
We have determined the conditions under which one dimensional biased quantum walks are recurrent. This together with the condition of being genuinely biased give rise to three different regions in the parameter space which we depict as a "phase diagram" in \fig{fig4}.
\begin{equation}gin{figure}[h]
\begin{equation}gin{center}
\includegraphics[width=0.6\textwidth]{fig4.eps}
\caption{"Phase diagram" of biased quantum walks on a line. The horizontal axis represents the length of the step to the right $r$ and the vertical axis shows the coin parameter $\rho$. The dotted line corresponds to the recurrence criterion (\ref{crit:rec}), while the squares represent the condition (\ref{rho:0}) on the zero mean value of the particle's position. The quantum walks in the white area are transient and genuine biased. In between the two curves (light gray area) we find quantum walks which are recurrent but still genuine biased. The quantum walks in the dark gray area are recurrent and for a particular choice of the initial state they can produce probability distribution with vanishing mean value.}
\label{fig4}
\end{center}
\end{figure}
The presented results allow for generalization to biased quantum walks in higher dimensions assuming we keep the coin operator
in a tensorial form. For non-factorizable coin operators in higher dimensions it remains an open question when they are recurrent or transient.
\ack
The financial support by MSM 6840770039, M\v SMT LC 06002, the Czech-Hungarian cooperation project (KONTAKT,CZ-10/2007) and by the Hungarian
Scientific Research Fund (T049234) is gratefully acknowledged.
\begin{equation}gin{appendix}
\section{Recurrence of classical biased random walk on a line}
\label{app:a}
Classical random walks on a line can be biased in two ways - the step in one direction is greater than in the other one and the probability of
the step to the right is different from the probability of the step to the left (see \fig{fig5}).
\begin{equation}gin{figure}[h]
\begin{equation}gin{center}
\includegraphics[width=0.6\textwidth]{fig5.eps}
\caption{Schematics of the biased random walk on a line. The particle can move to the right by a distance $r$ with the probability $p$.
The length of the step to the left is unity and the probability of this step is $1-p$.}
\label{fig5}
\end{center}
\end{figure}
Consider a random walk on a line such that the particle can make a
jump of length $r$ to the right with probability $p$ or make a unit
size step to the left with probability $1-p$. The random walk is
recurrent if and only if the probability to find the particle at the
origin at any time instant $t$ does not decays faster than $t^{-1}$.
This probability is easily found to be expressed by the binomial
expression
\begin{equation}gin{equation}
P_0(t) = (1-p)^{\frac{t r}{r+1}}p^{\frac{t}{r+1}}{t\choose \frac{t r}{r+1}}.
\end{equation}
With the help of the Stirling's formula
\begin{equation}gin{equation}
n! \approx \sqrt{2\pi n}\left(\frac{n}{e}\right)^n
\end{equation}
we find the asymptotic behaviour of the probability at the origin
\begin{equation}gin{equation}
P_0(t)\approx \frac{r+1}{\sqrt{2\pi r t}}\left[(1-p)^{\frac{r}{r+1}}p^{\frac{1}{r+1}}\frac{r+1}{r^\frac{r}{r+1}}\right]^t.
\end{equation}
The asymptotics of the probability $P_0(t)$ therefore depends on
the value of
\begin{equation}gin{equation}
q = (1-p)^{\frac{r}{r+1}}p^{\frac{1}{r+1}}\frac{r+1}{r^\frac{r}{r+1}}.
\end{equation}
Since $q\leq 1$ the probability $P_0(t)$ decays exponentially
unless the inequality is saturated. Hence, the random walk is
recurrent if and only if $q$ equals unity. This condition is
satisfied for
\begin{equation}gin{equation}
\label{rw:cond}
p = \frac{1}{r+1},
\end{equation}
i.e. the probability of the step to the right has to be inversely
proportional to the length of the step.
This result can be well understood from a different point of view, as we illustrate in \fig{fig6}.
The spreading of the probability distribution is diffusive, i.e.
$\sigma\sim\sqrt{t}$. The probability in the $\sigma$ neighborhood
of the mean value $\langle x\rangle$ behaves like $t^{-\frac{1}{2}}$
while outside this neighborhood the probability decays
exponentially. Therefore for the random walk to be recurrent the
origin must lie in this $\sigma$ neighborhood for all times $t$.
However, if the random walk is biased the mean value of the position
$\langle x\rangle$ varies linearly in time, thus it is a faster
process than the spreading of the probability distribution. In such
a case the origin would lie outside the $\sigma$ neighborhood of the mean
value after a finite number of steps leading to the exponential
asymptotic decay of the probability at the origin $P_0(t)$. Hence,
the random walk is recurrent if and only if the mean value of the
position equals zero. Since the individual steps are independent of
each other the mean value after $t$ steps is simply a $t$ multiple
of the mean value after single step, i.e.
\begin{equation}gin{equation}
\langle x (t)\rangle = t \langle x(1)\rangle = t\left[p(r+1)-1\right].
\end{equation}
We find that the mean value equals zero if and only if the condition (\ref{rw:cond})
holds.
\begin{equation}gin{figure}[h]
\begin{equation}gin{center}
\includegraphics[width=0.6\textwidth]{fig6.eps}
\caption{Spreading of the probability distribution versus the motion
of the mean value of a biased classical random walk on a line. While the
spreading is diffusive ($\sigma\sim\sqrt{t}$) the mean value
propagates with a constant velocity ($\langle x\rangle\sim t$). The
probability inside the $\sigma$ neighborhood of the mean value
$\langle x\rangle$ behaves like $t^{-\frac{1}{2}}$. On the other
hand, as we go away from the $\sigma$ neighbourhood the decay is exponential.
Hence, if the mean value $\langle x\rangle$ does not vanish the
origin of the walk leaves the $\sigma$ neighborhood of the mean
value. In such a case the probability at the origin decays
exponentially and the walk is transient.}
\label{fig6}
\end{center}
\end{figure}
| 3,029 | 16,635 |
en
|
train
|
0.40.4
|
\section{Mean value of the particle's position for a quantum walk on a line}
\label{app:b}
In this Appendix we find the explicit form of the position mean
value of the particle. With the help of the weak limit theorem
\cite{Grimmett} we express the mean value after $t$ steps in the
form
\begin{equation}gin{equation}
\left\langle \frac{x}{t}\right\rangle \approx \sum_{j=1}^2\int_{-\pi}^\pi\frac{dk}{2\pi}\ \omega_j'(k)\ \left(v_j(k),\psi\right)v_j(k),
\end{equation}
up to the corrections of the order $O(t^{-1})$. Here $v_j(k)$ are eigenvectors of the
unitary propagator $\widetilde{U}(k)$, $\omega_j'(k)$ are the
derivatives of the phases of the corresponding eigenvalues and
$\psi$ is the initial state expressed in (\ref{psi:init}). The
derivatives of the phases are given in (\ref{phase:der}). We express
the eigenvectors in the form
\begin{equation}gin{eqnarray}
\nonumber v_1(k) & = & n_1(\rho,k)\left(\sqrt{1-\rho}, -\sqrt{\rho} + e^{i(\omega_1(k)-rk)}\right)^T,\\
\nonumber v_2(k) & = & n_2(r,k)\left(\sqrt{1-\rho}, -\sqrt{\rho} + e^{i(\omega_2(k)-rk)}\right)^T.\\
\end{eqnarray}
The normalization factors of the eigenvectors read
\begin{equation}gin{eqnarray}
\nonumber n_1(u) & = & 2-2\sqrt{\rho}\cos\left(u-\arcsin\left[\sqrt{\rho}\sin u\right]\right),\\
\nonumber n_2(u) & = & 2+2\sqrt{\rho}\cos\left(u+\arcsin\left[\sqrt{\rho}\sin u\right]\right),\\
\end{eqnarray}
where we denote $u=\frac{k(r+1)}{2}$ to shorten the notation. The mean value is thus given by the following integral
\begin{equation}gin{equation}
\left\langle \frac{x}{t}\right\rangle \approx \int\limits_0^{(r+1)\pi}\frac{f(a,\varphi,\rho,r,u)du}{2(r+1)\pi \left[1 +\sqrt{\rho}\cos u_1\right] \left[1-\sqrt{\rho} \sin u_2\right]}+O(t^{-1}),
\end{equation}
where
\begin{equation}gin{equation}
u_1 = u + \arcsin(\sqrt{\rho}\sin u),\quad u_2 = u + \arccos(\sqrt{\rho}\sin u),
\end{equation}
and the numerator reads
\begin{equation}gin{eqnarray}
\nonumber f(a,\varphi,\rho,r,u) & = & (1-\rho) \left[r-1 + \rho \left(a + r (a-1)\right)\left(1+\cos(2u)\right)+\right.\\
\nonumber & & \left.+ \sqrt{a(1-a)}\sqrt{\rho(1-\rho)} (r+1) \left(\cos{\varphi}+\cos(\varphi+2u)\right)\right].\\
\end{eqnarray}
Performing the integrations we arrive at the result
\begin{equation}gin{eqnarray}
\nonumber \left\langle\frac{x}{t}\right\rangle & \approx & (1-\sqrt{1-\rho})(a(r+1)-1)+\\
\nonumber & & +\frac{\sqrt{a(1-a)}(1-\sqrt{1-\rho})(1-\rho)(r+1)\cos\varphi}{\sqrt{\rho(1-\rho)}}\\
& & +\frac{r-1}{2}\sqrt{1-\rho}+O(t^{-1}).
\end{eqnarray}
\end{appendix}
\section*{References}
\begin{equation}gin{thebibliography}{x}
\bibitem{overview}
Guillotin-Plantard N and Schott R 2006 {\it Dynamic Random Walks: Theory and Application} Elsevier Amsterdam
\bibitem{hughes}
Hughes B D 1995 {\it Random walks and random environments, Vol. 1: Random walks} Oxford University Press Oxford
\bibitem{bachelier}
Bachelier L 1900 {\it Ann. Sci. Ecole Norm. Super.} S\'er. 3 \textbf{17} 21
\bibitem{chandrasekhar:1943}
Chandrasekhar S 1943 {\it Rev. Mod. Phys.} \textbf{15} 1
\bibitem{aharonov}
Aharonov Y, Davidovich L and Zagury N 1993 {\it Phys. Rev.} A \textbf{48} 1687
\bibitem{konno:book}
Konno N 2008 in {\it Quantum Potential Theory}, Eds. M. Schurmann and U. Franz, Lecture Notes in Mathematics \textbf{1954} pp. 309 Springer-Verlag Berlin
\bibitem{kempe}
Aharonov D, Ambainis A, Kempe J and Vazirani U 2001 in {\it Proceedings of the 33th STOC} New York 50
\bibitem{shenvi:2003}
Shenvi N, Kempe J and Whaley K B 2003 {\it Phys. Rev.} A \textbf{67} 052307
\bibitem{aurel:2007}
Gabris A, Kiss T and Jex I 2007 {\it Phys. Rev.} A \textbf{76} 062315
\bibitem{santha}
Santha M 2008 in {\it Theory and Applications of Models of Computation}, Eds. M. Agrawal, D.Z. Du, Z.H. Duan and A. S. Li, Lecture Notes In Computer Science \textbf{4978} pp. 31 Springer-Verlag Berlin
\bibitem{kempe:2005}
Kempe J 2005 {\it Prob. Th. Rel. Fields} \textbf{133} (2) 215
\bibitem{krovi:2006a}
Krovi H and Brun T A 2006 {\it Phys. Rev.} A \textbf{73} 032341
\bibitem{krovi:2006b}
Krovi H and Brun T A 2006 {\it Phys. Rev.} A \textbf{74} 042334
\bibitem{magniez}
Magniez F, Nayak A, Richter P C and Santha M 2008 arXiv:0808.0084
\bibitem{nayak}
Ambainis A, Bach E, Nayak A, Vishwanath A and Watrous J 2001 in {\it Proceedings of the 33th STOC} New York 37
\bibitem{optical:galton}
Bouwmeester D, Marzoli I, Karman G P, Schleich W and Woerdman J P 2000 {\it Phys. Rev.} A \textbf{61} 013410
\bibitem{tregenna}
Tregenna B, Flanagan W, Maile R and Kendon V 2003 {\it New J. Phys.} \textbf{5} 83.1
\bibitem{wojcik}
Wojcik A, Luczak T, Kurzynski P, Grudka A and Bednarska M 2004 {\it Phys. Rev. Lett.} \textbf{93} 180601
\bibitem{knight}
Knight P L, Roldan E and Sipe J E 2003 {\it Phys. Rev.} A \textbf{68} 020301
\bibitem{carteret}
Carteret H A, Ismail M E H and Richmond B 2003 {\it J. Phys.} A \textbf{36} 8775
\bibitem{chandrashekar:2007}
Chandrashekar C M, Srikanth R and Laflamme R 2008 {\it Phys. Rev.} A \textbf{78} 052316
\bibitem{konno:2002}
Konno N 2002 {\it Quantum Inform. Compu.} \textbf{2} 578
\bibitem{1dloc1}
Inui N and Konno N 2005 {\it Physica} A \textbf{353} 133
\bibitem{1dloc}
Inui N, Konno N and Segawa E 2005 {\it Phys. Rev.} E \textbf{72} 056112
\bibitem{miyazaki}
Miyazaki T, Katori M and Konno N 2007 {\it Phys. Rev.} A \textbf{76} 012332
\bibitem{sato}
Sato M, Kobayashi N, Katori M and Konno N 2008 arXiv:0802.1997
\bibitem{polya}
P\'olya G 1921 {\it Mathematische Annalen} \textbf{84} 149
\bibitem{revesz}
R\'ev\'esz P 1990 {\it Random walk in random and non-random environments} World Scientific Singapore
\bibitem{domb:1954}
Domb C 1954 {\it Proc. Cambridge Philos. Soc.} \textbf{50} 586
\bibitem{montroll:1964}
Montroll E W 1964 in {\it Random Walks on Lattices}, edited by R. Bellman, Vol. \textbf{16} 193 American Mathematical Society Providence RI
\bibitem{prl}
\v Stefa\v n\'ak M, Jex I and Kiss T 2008 {\it Phys. Rev. Lett.} \textbf{100} 020501
\bibitem{pra}
\v Stefa\v n\'ak M, Kiss T and Jex I 2008 {\it Phys. Rev.} A \textbf{78} 032306
\bibitem{peres}
Peres A 1982 {\it Phys. Rev. Lett.} \textbf{49} 1118
\bibitem{chandra:recurrence}
Chandrashekar C M 2008 arXiv:0810.5592
\bibitem{statphase}
Wong R 2001 {\it Asymptotic Approximations of Integrals} SIAM Philadelphia
\bibitem{Grimmett}
Grimmett G, Janson S and Scudo P F 2004 {\it Phys. Rev.} E \textbf{69} 026119
\end{thebibliography}
\end{document}
| 2,849 | 16,635 |
en
|
train
|
0.41.0
|
\begin{document}
\title{ A Linear Scalarization Proximal Point Method for Quasiconvex Multiobjective Minimization }
\author{E. A. Papa Quiroz\thanks{ Mayor de San Marcos National University, Department of Mathematical Sciences, Lima, Per\'{u} and Federal University of Rio de Janeiro, Computing and Systems Engineering Department, post office box 68511,CEP 21945-970, Rio de Janeiro, Brazil([email protected]).}
\and{H. C. F. Apolinario\thanks{Federal University of Tocantins, Undergraduate Computation Sciences Course, ALC NO 14 (109 Norte) AV.NS.15 S/N , CEP 77001-090, Tel: +55 63 8481-5168; +55 63 3232-8027; FAX +55 63 3232-8020, Palmas, Brazil ([email protected]).}}\\
\and{K.D.V. Villacorta\thanks{Federal University of Paraíba, Campus V-Mangabeira, João Pessoa-Paraíba, Brazil. CEP: 58.055-000 }}
\and{P. R. Oliveira\thanks{Federal University of Rio de Janeiro, Computing and Systems Engineering Department, post office box 68511,CEP 21945-970, Rio de Janeiro, Brazil([email protected]).}}}
\date{\today}
\maketitle
\ \\[-0.5cm]
\begin{center}
{\bf Abstract}
\end{center}
In this paper we propose a linear scalarization proximal point algorithm for solving arbitrary lower semicontinuous quasiconvex multiobjective minimization problems. Under some natural assumptions and using the condition that the proximal parameters are bounded we prove the convergence of the sequence generated by the algorithm and when the objective functions are continuous, we prove the convergence to a generalized critical point. Furthermore, if each iteration minimize the proximal regularized function and the proximal parameters converges to zero we prove the convergence to a weak Pareto solution. In the continuously differentiable case, it is proved the global convergence of the sequence to a Pareto critical point and we introduce an inexact algorithm with the same convergence properties. We also analyze particular cases of the algorithm obtained finite convergence to a Pareto optimal point when the objective functions are convex and a sharp minimum condition is satisfied.
\\\\
\noindent{\bf Keywords:} Multiobjective minimization, lower semicontinuous quasiconvex functions, proximal point methods, Fejér convergence, Pareto-Clarke critical point, finite convergence.
\section{Introduction}
\noindent
In this work we consider the multiobjective minimization problem:
\begin{eqnarray}
\textrm{min}\lbrace F(x): x \in \mathbb{R}^n\rbrace
\label{prob}
\end{eqnarray}
where $F=(F_1,F_2,...,F_m): \mathbb{R}^n\longrightarrow \mathbb{R}^m\cup \{+ \infty \}^m$ is a lower semicontinuous and
quasiconvex vector function on the Euclidean space $ \mathbb{R}^n.$ The above notation means that each $F_i$ is an extended function, that is, $F_i:\mathbb{R}^n\longrightarrow \mathbb{R}^m\cup \{+ \infty \}$ .
The main motivation to study this problem are the consumer demand theory in economy, where the quasiconvexity of the objective vector function is a natural condition associated to diversification of the consumption, see Mas Colell et al. \cite{Colell}, and the quasiconvex optimization models in location theory, see \cite{Gromicho}.
Recently Apolinario et al. \cite{apo} has been introduced an exact linear scalarization proximal point algorithm to solve the above class of problems when the vector function $F$ is locally Lipschitz and $\textnormal{dom}(F)=\mathbb{R}^n$. The proposed iteration was the following:
given $p^{k} \in \mathbb{R}^n$, find $p^{k+1}\in \Omega_k=\left\{ x\in \mathbb{R}^n: F(x) \preceq F(p^k)\right\}$ such that:
$$
0 \in \partial^o\left( \left\langle F(.), z_k\right\rangle + \dfrac{\alpha_k}{2} \Vert\ .\ - p^k \Vert ^2 \right) (p^{k+1}) + \mathcal{N}_{\Omega_k}(p^{k+1})
$$
where ${\partial}^o$ is the Clarke subdifferential, see Subsection 2.5 of \cite{apo}, $\alpha_k > 0 $, $\left\{z_k\right\} \subset \mathbb{R}^m_+\backslash \left\{0\right\}$, $\left\|z_k\right\| = 1$ and $\mathcal{N}_{\Omega_k}(p^{k+1})$ the normal cone to $\Omega_k$ at $x^{k+1},$ see Definition \ref{normal} in Section \ref{Prelimin} of this paper. The authors proved, under some natural assumptions, that the sequence generated by the above algorithm is well defined and converges globally to a Pareto-Clarke critical point.
Unfortunately, the algorithm proposed in that paper can not be applied to a general class of proper lower semicontinuous quasiconvex functions, and thus can not be applied to solve constrained multiobjective problems nor continuous quasiconvex functions which are not locally Lipschitz. Moreover, for a future implementation and application for example to costly improving behaviors of strongly averse agents in economy (see Sections 5 and 6 of Bento et al. \cite{Bento}), that paper did not provide an inexact version of the proposed algorithm .
Thus we have two motivations in the present paper: the first motivation is to extend the convergence properties of the linear scalarization proximal point method introduced in \cite{apo} to solve more general, probabily constrained, quasiconvex multiobjective problems of the form (\ref{prob}) and the second ones is to introduce an inexact algorithm when $F$ is continuously differentiable on $\mathbb{R}^m$.
The main iteration of the proposed algorithm is: Given $x^k,$ find $x^{k+1} $ such that
\begin{eqnarray}
0 \in \hat{\partial}\left( \left\langle F(.), z_k\right\rangle + \dfrac{\alpha_k}{2} \Vert\ .\ - x^k \Vert ^2 + \delta_{\Omega_k}(.) \right) (x^{k+1})
\label{subdiferencial3i}
\end{eqnarray}
where $\hat{\partial}$ is the Fréchet subdifferential, see Subsection \ref{frechet}, \ $\Omega_k= \left\{ x\in \mathbb{R}^n: F(x) \preceq F(x^k)\right\}$, $\alpha_k > 0 $, $\left\{z_k\right\} \subset \mathbb{R}^m_+\backslash \left\{0\right\}$ and $\left\|z_k\right\| = 1$.
Some works related to this paper are the following:
\begin{itemize}
\item Bento et al. \cite{Bento} introduced the nonlinear scalarized proximal iteration:
$$
y^{k+1}\in \arg \min \left\{f\left( F(x)+ \delta_{\Omega_k}(x)e+ \dfrac{\alpha_k}{2} \Vert\ .\ - y^k \Vert ^2e \right): x\in \mathbb{R}^n \right\}
$$
where $f:\mathbb{R}^n\longrightarrow \mathbb{R}$ is a function defined by $f(y):=\max_{i\in I}\{ \langle y,e_i \rangle \}$
with $e_i$ is the canonical base of the space $\mathbb{R}^n,$ $\Omega_k= \left\{ x\in \mathbb{R}^n: F(x) \preceq F(y^k)\right\}$ and $e=(1,1,...,1)\in \mathbb{R}^n.$ Assuming that $F:\mathbb{R}^n\longrightarrow \mathbb{R}^m$ is quasiconvex and continuously differentiable and under some natural assumptions the authors proved that the sequence $\{y^k\}$ converges to a Pareto Critical point of $F.$ Furthermore, assuming that $F$ is convex, the weak Pareto optimal set is weak sharp for the multiobjective problem and that the sequence is generated by the following unconstrained iteration
$$
y^{k+1}:=\arg \min \left\{f(F(x)) + \dfrac{\alpha_k}{2} \Vert\ x\ - y^k \Vert ^2:x\in \mathbb{R}^n \right\}
$$
then the above iteration obtain a Pareto optimal point after a finite number de iterations.
The difference between our work and the paper of Bento et al., \cite{Bento}, is that in the present paper we consider a linear scalararization of $F$ instead of a nonlinear ones proposed in \cite{Bento}, another difference is that our assumptions are more weak, in particular, we obtain convergence results for nondifferentiable quasiconvex functions.
\item Makela et al., \cite{makela}, developed a multiobjective proximal bundle method for nonsmooth optimization where the objective functions are locally Lipschitz (not necessarily smooth nor convex). The proximal method is not directly based on employing any scalarizing function but based on a improvement function $H:\mathbb{R}^n\times \mathbb{R}^n \longrightarrow \mathbb{R}$ defined by $H(x,y)=\max \{ F_i(x)-F_i(y),g_j(x):i=1,...,m, j=1,...,r \}$ with $\textnormal{dom} F=\{x\in \mathbb{R}^n: g_j(x)\geq 0, j=1,...,r\}.$ If $F_i$ and $g_j$ are pseudoconvex and weakly semismooth functions and certain constraint qualification is valid, the authors proved that any accumulation point of the sequence is a weak Pareto solution and without the assumption of pseudoconvex, they obtained that any accumulation point is a substationary point, that is, $0\in \partial H(\bar x, \bar x),$ where $\bar x$ is an accumulation point.
\item Chuong et al., \cite{chuong}, developed three algorithms of the so-called hybrid approximate proximal type to find Pareto optimal points for general class of convex constrained problems of vector optimization in finite and infinite dimensional spaces, that is, $\min_C \{F(x): x\in \Omega\},$ where $C$ is a closed convex and pointed cone and the minimization is understood with respect to the ordering relation given by $y\preceq_{C}x$ if and only if $x-y\in C$. Assuming that the set $\left(F(x^0) - C\right) \cap F(\Omega)$ is $C$ - quasi-complete for $\Omega,$ that is, for any sequence $\{u_l\}\subset \Omega$ with $u_0=x_0$ such that $F(u_{l+1})\preceq_{C}F(u_l)$ there exists $u\in VI(\Omega, A)$ satisfying $F(u)\preceq_{C}F(u_l),$ for every $l\in \mathbb{N};$ and the assumption that $F$ is $C^{+}-$ uniformly semicontinuous on $\Omega,$ the authors proved the convergence of the sequence generates by its algorithm.
\end{itemize}
Under the assumption that $F$ is a proper lower semicontinuous quasiconvex vector function and the assumption that the set $\left(F(x^0) - \mathbb{R}^m_+\right)\cap F(\mathbb{R}^n)$ is $\mathbb{R}^m_+$ - complete we prove the global convergence of the sequence $\{x^k\},$ generated by (\ref{subdiferencial3i}), to the set
$$E = \left\{x \in \mathbb{R}^n: F\left(x\right)\preceq F\left(x^k\right),\ \ \forall\ k \in \mathbb{N}\right\}.$$
Additionally, if $F:\mathbb{R}^n\longrightarrow \mathbb{R}^m $ is continuous, and $0<\alpha_k<\bar{\alpha},$ for some $\bar{\alpha}>0,$ we prove that
$\lim \limits_{k\rightarrow +\infty}g^{k} = 0,$ where $g^k \in \hat{\partial}\left( \left\langle F(.), z_k\right\rangle + \delta_{\Omega_k}\right)(x^{k+1}).$ In the particular case when $ \lim \limits_{k\rightarrow +\infty}\alpha_k= 0$ and the iterations are given by
\begin{eqnarray}
x^{k+1}\in \textnormal{arg min} \left\{\left\langle F(x), z_k\right\rangle+\frac{\alpha_k}{2}\left\|x - x^k\right\|^2 : x\in\Omega_k\right\},
\label{recursao0i}
\end{eqnarray}
then the sequence $\lbrace x^k\rbrace$ converges to a weak pareto solution of the problem $(\ref{prob})$.
When the vector function $F: \mathbb{R}^n\longrightarrow \mathbb{R}^m$ is continuously differentiable and $0<\alpha_k<\bar{\alpha},$ for some $\bar{\alpha}>0,$ we prove that the sequence $\{x^k\},$ generated by (\ref{subdiferencial3i}), converges to a Pareto critical point of the problem (\ref{prob}). Then, we introduce an inexact proximal algorithm given by
\begin{equation}
0 \in \hat{\partial}_{\epsilon_k} \left( \langle F(x), z_k \rangle \right) (x) + \alpha_k\left(x - x^k\right) + \mathcal{N}_{\Omega_k}(x),
\label{diferenciali}
\end{equation}
\begin{equation}
\label{deltai}
\displaystyle \sum_{k=1}^{\infty} \delta_k < \infty,
\end{equation}
where $\delta_k = \textnormal{max}\left\lbrace \dfrac{\varepsilon_k}{\alpha_k}, \dfrac{\Vert \nu_k\Vert}{\alpha_k}\right\rbrace,$ $\varepsilon_k\geq 0,$ and $\hat{\partial}_{\varepsilon_k}$ is the Fréchet $\varepsilon_k$-subdifferential.
We prove the convergence of $\{x^k\},$ generated by (\ref{diferenciali}) and (\ref{deltai}) to a Pareto critical point of the problem (\ref{prob}).
We also analyze some conditions to obtain finite convergence of a particular case of the proposed algorithm.
The paper is organized as follows: In Section 2 we recall some concepts and basic results on multiobjective optimization, descent direction, scalar representation, quasiconvex and convex functions, Fr\'echet and Limiting subdiferential, $\epsilon-$Subdifferential and Fej\'{e}r convergence. In Section 3 we present the problem and we give an example of a quasiconvex model in demand theory. In Section 4 we introduce an exact algorithm and analyze its convergence. In Section 5 we present an inexact algorithm for the differentiable case and analyze its convergence. In Section 6, we introduce an inexact algorithm for nonsmooth proper lower semicontinuous convex multiobjective minimization and using some concepts of weak sharp minimum we prove the convergence of the iterations in a finite number of steps to a Pareto optimal point. In Section 7 give a numerical example of the algorithm and in Section 8 we give our conclusions.
| 3,914 | 30,721 |
en
|
train
|
0.41.1
|
\section{Preliminaries}
\label{Prelimin}
In this section, we present some basic concepts and results that are of fundamental importance for the development of our work. These facts can be found, for example, in Hadjisavvas \cite{Had}, Mordukhovich \cite{Mordukhovich} and, Rockafellar and Wets \cite{Rockafellar}.
\subsection{Definitions, notations and some basic results}
Along this paper $ \mathbb{R}^n$ denotes an Euclidean space, that is, a real vectorial space with the canonical inner product $\langle x,y\rangle=\sum\limits_{i=1}^{n} x_iy_i$ and the norm given by $||x||=\sqrt{\langle x, x\rangle }$.\\
Given a function {\small $f :\mathbb{R}^n\longrightarrow \mathbb{R}\cup\left\{+\infty\right\}$}, we
denote by $\textnormal{dom}(f)= \left\{x \in \mathbb{R}^n: f(x) < + \infty \right\},$ the {\it effective domain } of $f$.
If $\textnormal{dom}(f) \neq \emptyset$, $f $ is called proper.
If {\footnotesize $\lim \limits_{\left\|x\right\|\rightarrow +\infty}f(x) = +\infty$}, $f$ is called coercive. We denote by arg min $\left\{f(x): x \in \mathbb{R}^n\right\}$ the set of minimizer of the function $f$ and by $f * $, the optimal value of problem: $\min \left\{f(x): x \in \mathbb{R}^n\right\},$ if it exists.
The \ function \ $f$ is {\it lower semicontinuous} at $\bar{x}$ if for all sequence $\left\{x_k\right\}_{k \in \mathbb{N}} $ such that $\lim \limits_{k \rightarrow +\infty}x_k = \bar{x}$ we obtain that $f(\bar{x}) \leq \liminf \limits_{k \rightarrow +\infty}f(x_k)$.\\
The next result ensures that the set of minimizers of a function, under some assumptions, is nonempty.
\begin{proposicao}{\bf (Rockafellar and Wets \cite{Rockafellar}, Theorem 1.9)}\\
Suppose that {\small $f:\mathbb{R}^n\longrightarrow \mathbb{R}\cup\left\{+\infty\right\}$} is {\it proper, lower semicontinuous} and coercive, then the optimal value $ f^*$\ is finite and the set $\textnormal{arg min}$ $\left\{f(x): x \in \mathbb{R}^n\right\}$ is nonempty and compact.
\label{coercivaesemicont}
\end{proposicao}
\begin{Def}
Let $D \subset \mathbb{R}^n$ and $\bar{x} \in D$. The normal cone to $D$ at $\bar{x} \in D$ is given by $\mathcal{N}_{D}(\bar{x}) = \left\{v \in \mathbb{R}^n: \langle v, x - \bar{x}\rangle \leq 0, \forall \ x \in D\right\}$.
\label{normal}
\end{Def}
It follows an important result that involves sequences of non-negative numbers which will be useful in Section 5.
\begin{lema}
Let $\{w_k\}$, $\{p_k\}$ and $\{q_k\}$ sequences of non-negative real numbers. If
\begin{equation*}
w_{k+1} \leq \left( 1 + p_k\right)w_k + q_k, \ \ \ \ \displaystyle \sum_{i=1}^{\infty} p_k < +\infty \ \ \textnormal{and} \ \ \displaystyle \sum_{i=1}^{\infty} q_k < +\infty,
\end{equation*}
then the sequence $\{w_k\}$ is convergent.
\label{p}
\end{lema}
\begin{proof}
See Polyak \cite{Polyak}, Lema 2.2.2.
\end{proof}
\subsection{Multiobjective optimization}
In this subsection we present some properties and notation on multiobjective optimization. Those basic facts can be seen, for example, in Miettinen \cite{Kaisa} and Luc \cite{Luc}.\\
Throughout this paper we consider the cone $\mathbb{R}^m_+ = \{ y\in \mathbb{R}^m : y_i\geq0, \forall \ i = 1, ... , m \}$, which induce a partial order $\preceq$ in $\mathbb{R}^m$ given by, for $y,y'\in \mathbb{R}^m$,
$y\ \preceq\ y'$ if, and only if, $ y'\ - \ y$ $ \in \mathbb{R}^m_+$, this means that $ y_i \leq \ y'_i,$ for all $ i= 1,2,...,m $ . Given $ \mathbb{R}^m_{++}= \{ y\in \mathbb{R}^m : y_i>0, \forall \ i = 1, ... , m \}$ the above relation induce the following one $\prec$, induced by the interior of this cone, given by, $y\ \prec\ y'$, if, and only if, $ y'\ - \ y$ $ \in \mathbb{R}^m_{++}$, this means that $ y_i < \ y'_i$ for all $ i= 1,2,...,m$. Those partial orders establish a class of problems known in the literature as Multiobjective Optimization.\\ \\
Let us consider the unconstrained multiobjective optimization problem (MOP) :
\begin{eqnarray}
\textrm{min} \left\{G(x): x \in \mathbb{R}^n \right\}
\label{POM}
\end{eqnarray}
where $G:\mathbb{R}^n\longrightarrow \mathbb{R}^m\cup \{+ \infty \}^m$, with $G = \left(G_1, G_2, ... , G_m\right)$ and $G_i:\mathbb{R}^n \longrightarrow \mathbb{R}, \forall i=1,...,m$.
\begin{Def} {\bf (Miettinen \cite{Kaisa}, Definition 2.2.1)}
A point $x^* \in \mathbb{R}^n$ is a Pareto optimal point or Pareto solution of the problem $\left(\ref{POM}\right)$, if there does not exist $x \in \mathbb{R}^n $ such that $ G_{i}(x) \leq G_{i}(x^*)$, for all $i \in \left\{1,...,m\right\}$ and $ G_{j}(x) < G_{j}(x^*)$, for at least one index $ j \in \left\{1,...,m\right\}$ .
\end{Def}
\begin{Def}{\bf (Miettinen \cite{Kaisa},Definition 2.5.1)}
A point $x^* \in \mathbb{R}^n$ is a weak Pareto solution of the problem $\left(\ref{POM}\right)$, if there does not exist $x \in \mathbb{R}^n $ such that $ G_{i}(x) < G_{i}(x^*)$, for all $i \in \left\{1,...,m\right\}$.
\end{Def}
We denote by arg min$\left\{G(x):x\in \mathbb{R}^n \right\}$ and by arg min$_w$ $\left\{G(x):x\in \mathbb{R}^n \right\}$ the set of Pareto solutions and weak Pareto solutions to the problem $\left(\ref{POM}\right)$, respectively. It is easy to check that\\ arg min$\left\{G(x):x\in \mathbb{R}^n \right\} \subset$ arg min$_w$ $\left\{G(x):x\in \mathbb{R}^n \right\}$.
\subsection{Pareto critical point and descent direction}
Let $G:\mathbb{R}^n\longrightarrow \mathbb{R}^m$ be a differentiable function and $x \in \mathbb{R}^n$, the jacobian of $G$ at $x$, denoted by $JG(x)$, is a matrix of order $m \times n$ whose entries are defined by $\left(JG(x)\right)_{i,j} = \frac{\partial G_i}{\partial x_j}(x)$. We may represent it by,
\begin{center}
$JG\left(x\right) := \left[\nabla G_1(x) \nabla G_2(x)... \nabla G_m(x) \right]^T$, $x \in \mathbb{R}^n$.
\end{center}
The image of the jacobian of $G$ at $x$ we denote by
\begin{center}
$\footnotesize{Im \left(JG\left(x\right)\right) := \lbrace JG\left(x\right)v = \left(\langle \nabla G_1(x) , v\rangle, \langle \nabla G_2(x) , v\rangle, ..., \langle \nabla G_m(x) , v\rangle\right): v \in \mathbb{R}^n \rbrace }$.
\end{center}
A necessary but not sufficient first order optimality condition for the problem
$(\ref {POM})$ at $x \in \mathbb {R}^n $, is
\begin{eqnarray}
Im \left(JG\left(x\right)\right)\cap\left(-\mathbb{R}^m_{++}\right)=\emptyset.
\label{cond}
\end{eqnarray}
Equivalently, $\forall \ v \in \mathbb{R}^n$, there exists $i_0 = i_0(v) \in \lbrace 1,...,m\rbrace$ such that
\begin{center}
$\langle \nabla G_{i_0}(x) , v \rangle \geq 0$.
\end{center}
\begin{Def}
Let $G:\mathbb{R}^n\longrightarrow \mathbb{R}^m$ be a differentiable function. A point $x^* \in \mathbb{R}^n$ satisfying $(\ref {cond})$ is called a Pareto critical point.
\end{Def}
Follows from the previous definition, if a point $x$ is not Pareto critical point, then there exists a direction
$v \in \mathbb {R}^ n$ satisfying
\begin{center}
$JG\left(x\right)v \in \left(-\mathbb{R}^m_{++}\right)$,
\end{center}
i.e, $\langle \nabla G_i( x ) , v \rangle < 0, \ \forall \ i \in \lbrace 1,..., m \rbrace$. As $G$ is continuously differentiable, then
\begin{center}
$\displaystyle \lim_{t \rightarrow 0}\dfrac{G_i(x + tv) - G_i(x)}{t}= \langle \nabla G_i(x) ,v\rangle < 0, \ \forall \ i \in \lbrace 1,..., m \rbrace $.
\end{center}
This implies that $v$ is a {\it descent direction} for the function $G_i$, i.e, there exists $\varepsilon > 0 $, such that
\begin{center}
$G_i(x + tv) < G_i(x), \forall \ t \in (0 , \varepsilon ], \forall \ i \in \lbrace 1,..., m \rbrace $.
\end{center}
Therefore, $v$ is a {\it descent direction} for $G$ at $x$, i.e, there exists $ \varepsilon > 0 $ such that
\begin{center}
$ G(x + tv) \prec G(x), \ \forall \ t \in (0 , \varepsilon]$.
\end{center}
\subsection{Scalar representation}
In this subsection we present a useful technique in multiobjective optimization which allows to replace the original optimization problem into a scalar optimization problem or a family of scalar problems.
\begin{Def}{\bf (Luc \cite{Luc}, Definição 2.1)}
A function
$f:\mathbb{R}^n\longrightarrow \mathbb{R}\cup \{+ \infty \}$ is said to be a strict scalar representation of a map $F:\mathbb{R}^n\longrightarrow \mathbb{R}^m\cup \{+ \infty \}^m$ when given $x,\bar{x}\in \mathbb{R}^n :$
\begin{center}
$F(x)\preceq F(\bar{x}) \Longrightarrow f(x)\leq f(\bar{x})$\ \ and \ \
$F(x)\prec F(\bar{x}) \Longrightarrow f(x)<f(\bar{x}).$
\end{center}
Furthermore, we say that $f$ is a weak scalar representation of $F$ if
\begin{center}
$F(x)\prec F(\bar{x})\Longrightarrow f(x)<f(\bar{x}).$
\end{center}
\label{escalarizacao}
\end{Def}
\begin{proposicao}
\label{rep}
Let $f:\mathbb{R}^n\longrightarrow \mathbb{R}\cup \{+ \infty \}$ be a proper function. Then $f$ is a strict scalar representation of $F$ if, and only if, there exists a strictly increasing function $g:F\left(\mathbb{R}^n\right)\longrightarrow \mathbb{R}$ such that $f = g \circ F.$
\end{proposicao}
\begin{proof}
See Luc \cite{Luc}, Proposition 2.3.
\end{proof}
\begin{proposicao}
\label{inclusao}
Let $f:\mathbb{R}^n\longrightarrow \mathbb{R}\cup \{+ \infty \}$ be a weak scalar representation of a vector function $F:\mathbb{R}^n\longrightarrow \mathbb{R}^m\cup \{+ \infty \}^m$ and $\textnormal{argmin}\left\{f(x):x\in \mathbb{R}^n\right\}$ the set of minimizer points of $f$. Then, we have
\begin{equation*}
\textnormal{argmin}\left\{f(x): x\in \mathbb{R}^n\right\}\subseteq \textnormal{argmin}_w \{F(x):x\in \mathbb{R}^n\}.
\end{equation*}
\end{proposicao}
\begin{proof}
It is immediate.
\end{proof}
| 3,551 | 30,721 |
en
|
train
|
0.41.2
|
\subsection{Scalar representation}
In this subsection we present a useful technique in multiobjective optimization which allows to replace the original optimization problem into a scalar optimization problem or a family of scalar problems.
\begin{Def}{\bf (Luc \cite{Luc}, Definição 2.1)}
A function
$f:\mathbb{R}^n\longrightarrow \mathbb{R}\cup \{+ \infty \}$ is said to be a strict scalar representation of a map $F:\mathbb{R}^n\longrightarrow \mathbb{R}^m\cup \{+ \infty \}^m$ when given $x,\bar{x}\in \mathbb{R}^n :$
\begin{center}
$F(x)\preceq F(\bar{x}) \Longrightarrow f(x)\leq f(\bar{x})$\ \ and \ \
$F(x)\prec F(\bar{x}) \Longrightarrow f(x)<f(\bar{x}).$
\end{center}
Furthermore, we say that $f$ is a weak scalar representation of $F$ if
\begin{center}
$F(x)\prec F(\bar{x})\Longrightarrow f(x)<f(\bar{x}).$
\end{center}
\label{escalarizacao}
\end{Def}
\begin{proposicao}
\label{rep}
Let $f:\mathbb{R}^n\longrightarrow \mathbb{R}\cup \{+ \infty \}$ be a proper function. Then $f$ is a strict scalar representation of $F$ if, and only if, there exists a strictly increasing function $g:F\left(\mathbb{R}^n\right)\longrightarrow \mathbb{R}$ such that $f = g \circ F.$
\end{proposicao}
\begin{proof}
See Luc \cite{Luc}, Proposition 2.3.
\end{proof}
\begin{proposicao}
\label{inclusao}
Let $f:\mathbb{R}^n\longrightarrow \mathbb{R}\cup \{+ \infty \}$ be a weak scalar representation of a vector function $F:\mathbb{R}^n\longrightarrow \mathbb{R}^m\cup \{+ \infty \}^m$ and $\textnormal{argmin}\left\{f(x):x\in \mathbb{R}^n\right\}$ the set of minimizer points of $f$. Then, we have
\begin{equation*}
\textnormal{argmin}\left\{f(x): x\in \mathbb{R}^n\right\}\subseteq \textnormal{argmin}_w \{F(x):x\in \mathbb{R}^n\}.
\end{equation*}
\end{proposicao}
\begin{proof}
It is immediate.
\end{proof}
\subsection{Quasiconvex and Convex Functions}
In this subsection we present the concept and characterization of quasiconvex functions and quasiconvex multiobjective function. This theory can be found in Bazaraa et al. \cite{Bazaraa}, Luc \cite{Luc}, Mangasarian \cite{Mangasarian}, and references therein.
\begin{Def}
Let $f:\mathbb{R}^n\longrightarrow \mathbb{R} \cup \{+ \infty \}$ be a proper function. Then, f is
called quasiconvex if for all $x,y\in \mathbb{R}^n$, and for all $ t \in \left[0,1\right]$, it holds
that $f(tx + (1-t) y)\leq \textnormal{max}\left\{f(x),f(y)\right\}$.
\end{Def}
\begin{Def}
Let $f:\mathbb{R}^n\longrightarrow \mathbb{R} \cup \{+ \infty \}$ be a proper function. Then, f is
called convex if for all $x,y\in \mathbb{R}^n$, and for all $ t \in \left[0,1\right]$, it holds
that $f(tx + (1-t) y)\leq tf(x) + (1 - t)f(y)$.
\end{Def}
Observe that if $f$ is a quasiconvex function then $\textnormal{dom}(f)$ is a convex set. On the other hand, while a convex function can be characterized by the convexity of its epigraph, a quasiconvex function can
be characterized by the convexity of the lower level sets:
\begin{Def}
Let \ \ $F= (F_1,...,F_m):\mathbb{R}^n\longrightarrow \mathbb{R}^m\cup \{+ \infty \}^m$ be a function, then $F$ is $\mathbb{R}^m_+$ - quasiconvex if every component function of $F$, $F_i: \mathbb{R}^n\longrightarrow \mathbb{R}\cup \{+ \infty \}$, is quasiconvex.
\end{Def}
\begin{Def}
Let \ \ $F= (F_1,...,F_m):\mathbb{R}^n\longrightarrow \mathbb{R}^m\cup \{+ \infty \}^m$ be a function, then $F$ is $\mathbb{R}^m_+$ - convex if every component function of $F$, $F_i: \mathbb{R}^n\longrightarrow \mathbb{R}\cup \{+ \infty \}$, is convex.
\end{Def}
\subsection{Fréchet and Limiting Subdifferentials}
\label{frechet}
\begin{Def}
Let $f: \mathbb{R}^n \rightarrow \mathbb{R} \cup \{ +\infty \}$ be a proper function.
\begin{enumerate}
\item [(a)]For each $x \in \textnormal{dom}(f)$, the set of regular subgradients (also called Fréchet subdifferential) of $f$ at $x$, denoted by $\hat{\partial}f(x)$, is the set of vectors $v \in \mathbb{R}^n$ such that
\begin{center}
$f(y) \geq f(x) + \left\langle v,y-x\right\rangle + o(\left\|y - x\right\|)$, where $\lim \limits_{y \rightarrow x}\frac{o(\left\|y - x\right\|)}{\left\|y - x\right\|} =0$.
\end{center}
Or equivalently, $\hat{\partial}f (x) := \left\{ v \in \mathbb{R}^n : \liminf \limits_{y\neq x,\ y \rightarrow x} \dfrac{f(y)- f(x)- \langle v , y - x\rangle}{\lVert y - x \rVert} \geq 0 \right \}$.\\ If $x \notin \textnormal{dom}(f)$ then $\hat{\partial}f(x) = \emptyset$.
\item [(b)]The set of general subgradients (also called limiting subdifferential) $f$ at $x \in \mathbb{R}^n$, denoted by $\partial f(x)$, is defined as follows:
\begin{center}
$\partial f(x) := \left\{ v \in \mathbb{R}^n : \exists\ x_l \rightarrow x, \ \ f(x_l) \rightarrow f(x), \ \ v_l \in \hat{\partial} f(x_l)\ \textnormal{and}\ v_l \rightarrow v \right \}$.
\end{center}
\end{enumerate}
\label{fech}
\end{Def}
\begin{proposicao}{\bf (Fermat’s rule generalized)}
If a proper function $f: \mathbb{R}^n \rightarrow \mathbb{R} \cup \{+ \infty \}$ has a local minimum at $\bar{x} \in \textnormal{dom}(f)$, then $0\in \hat{\partial} f\left(\bar{x}\right)$.
\label{otimo}
\end{proposicao}
\begin{proof}
See Rockafellar and Wets \cite{Rockafellar}, Theorem 10.1.
\end{proof}
\begin{proposicao}
Let $f: \mathbb{R}^n \rightarrow \mathbb{R} \cup \{+ \infty \}$ be a proper function. Then, the following properties are true
\begin{enumerate}
\item[(i)]$\hat{\partial}f(x) \subset \partial f(x)$, for all $x \in \mathbb{R}^n$.
\item[(ii)]If $f$ is differentiable at $\bar{x}$ then $\hat{\partial}f(\bar{x}) = \{\nabla f (\bar{x})\}$, so $\nabla f (\bar{x})\in \partial f(\bar{x})$.
\item[(iii)] If $f$ is continuously differentiable in a neighborhood of $x$, then $\hat{\partial}f(x) = \partial f(x) = \{\nabla f (x)\}$.
\item[(iv)] If \ $ g = f + h $ with $f$ finite at $\bar{x}$ and $h$ is continuously differentiable in a neighborhood of $\bar{x}$, then $\hat{\partial}g(\bar{x}) = \hat{\partial}f(\bar{x}) + \nabla h(\bar{x})$ and\
$\partial g(\bar{x}) = \partial f(\bar{x}) + \nabla h(\bar{x})$.
\end{enumerate}
\label{somafinita}
\end{proposicao}
\begin{proof}
See Rockafellar and Wets \cite{Rockafellar}, Exercise $8.8$, page $304$.
\end{proof}
\subsection{{\bf$\varepsilon$}-Subdiffential}
We present some important concepts and results on $\varepsilon$-subdifferential. The theory of these facts can be found, for example, in Jofre et al. \cite{Jofre} and Rockafellar and Wets \cite {Rockafellar}.
\begin{Def}
Let $f: \mathbb{R}^n \rightarrow \mathbb{R} \cup \{+ \infty \}$ be a proper lower semicontinuous function and let $\varepsilon$ be an arbitrary nonnegative real number. The Fréchet $\varepsilon$-subdifferential of $f$ at $x \in \textnormal{dom}(f)$ is defined by
\begin{eqnarray}
\hat{\partial}_{\varepsilon}f (x) := \left\{ x^* \in \mathbb{R}^n : \liminf \limits_{\lVert h \rVert \rightarrow 0} \dfrac{f(x + h)- f(x)- \langle x^* , h\rangle}{\lVert h \rVert} \geq - \varepsilon \right \}
\label{frechet1}
\end{eqnarray}
\label{frechet3}
\end{Def}
\begin{obs}
When $\varepsilon = 0$, $(\ref{frechet1})$ reduces to the well known Fréchet subdifferential, wich is denoted by $\hat{\partial}f(x)$, according to \textnormal{Definition} $\ref{fech}$. More precisely,
\begin{center}
$x^* \in \hat{\partial}f(x)$, if and only if, for each $\eta > 0 $ there exists $\delta > 0$ such that\\
$\langle x^* , y - x\rangle \leq f(y) - f(x) + \eta\Vert y - x \Vert$, for all $y \in x + \delta \textnormal{B}$,
\end{center}
where $B$ is the closed unit ball in $\mathbb{R}^n$ centered at zero. Therefore $\hat{\partial}f(x)=\hat{\partial}_{0}f(x)\subset\hat{\partial}_{\varepsilon}f(x).$
\label{frechet2}
\end{obs}
From Definition 5.1 of Treiman, \cite{Treiman},
\begin{center}
$x^* \in \hat{\partial}_{\epsilon}f (x)\Leftrightarrow x^* \in \hat{\partial}(f + \epsilon\Vert . - x\Vert)(x)$.
\end{center}
Equivalently, $x^* \in \hat{\partial}_{\epsilon}f (x)$, if and only if, for each $\eta > 0$, there exists $\delta > 0$ such that
\begin{center}
$\langle x^* , y - x\rangle \leq f(y) - f(x) + (\epsilon + \eta)\Vert y - x \Vert$, for all $y \in x + \delta \textnormal{B}$.
\end{center}
We now defined a new kind of approximate subdifferential.
\begin{Def}
The limiting Fréchet $\varepsilon$-subdifferential of $f$ at $x \in \textnormal{dom} (f)$ is defined by
\begin{eqnarray}
\partial_\varepsilon f(x) := \limsup \limits_{y \stackrel{f}{\longrightarrow} x} \hat{\partial}_{\varepsilon}f (y)
\end{eqnarray}
where $$\limsup \limits_{y \stackrel{f}{\longrightarrow} x} \hat{\partial}_{\varepsilon}f (y):=\lbrace x^* \in \mathbb{R}^n: \exists\ x_l \longrightarrow x, f(x_l)\longrightarrow f(x), x^*_l \longrightarrow x^* \ \textnormal{with}\ x^*_l \in \hat{\partial}_{\varepsilon}f (x_l)\, \rbrace$$
\end{Def}
In the case where $f$ is continuously differentiable, the limiting Fréchet $\varepsilon$-subdifferential takes a very simple form, according to the following proposition
\begin{proposicao}
Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be a continuously differentiable function at $x$ with derivative
$\nabla f (x)$. Then
\begin{center}
$\partial_\varepsilon f(x) = \nabla f (x) + \varepsilon B$.
\end{center}
\label{fdif}
\end{proposicao}
\begin{proof}
See Jofré et al., \cite{Jofre}, Proposition 2.8.
\end{proof}
\subsection{Fejér convergence}
\begin{Def}
A seguence $\left\{y_k\right\} \subset \mathbb{R}^n$ is said to be Fejér convergent to a set $U\subseteq \mathbb{R}^n$ if,
$\left\|y_{k+1} - u \right\|\leq\left\|y_k - u\right\|, \forall \ k \in \mathbb{N},\ \forall \ u \in U$.
\end{Def}
The following result on Fejér convergence is well known.
\begin{lema}
If $\left\{y_k\right\}\subset \mathbb{R}^n$ is Fejér convergent to some set $U\neq \emptyset$, then:
\begin{enumerate}
\item [(i)]The sequence $\left\{y_k\right\}$ is bounded.
\item [(ii)]If an accumulation point $y$ of $\left\{y_k\right\}$ belongs to $ U$, then $\lim \limits_{k\rightarrow +\infty}y_k = y$.
\end{enumerate}
\label{fejerlim1}
\end{lema}
\begin{proof}
See Schott \cite{Schott}, Theorem $2.7$.
\end{proof}
| 3,700 | 30,721 |
en
|
train
|
0.41.3
|
\section{The Problem}
We are interested in solving the multiobjective optimization problem (MOP):
\begin{eqnarray}
\textrm{min}\lbrace F(x): x \in \mathbb{R}^n\rbrace
\label{pom3}
\end{eqnarray}
where $F=\left( F_1, F_2,..., F_m\right): \mathbb{R}^n\longrightarrow \mathbb{R}^m\cup \{+ \infty \}^m$ is a vector function satisfying the following assumption:
\begin{description}
\item [$\bf (C_{1.1})$] $F$ is a proper lower semicontinuous vector function on $\mathbb{R}^n$, i.e, each $F_i:\mathbb{R}^n\longrightarrow \mathbb{R}^m\cup \{+ \infty \}$, $i=1,...,m$, is a proper lower semicontinuous function.
\item [$\bf (C_{1.2})$] $0 \preceq F.$
\end{description}
\subsection{A quasiconvex model in demand theory}
\noindent
Let $n$ be a finite number of consumer goods. A consumer is an agent who must choose how much to consume of each good. An ordered set of numbers representing the amounts consumed of each good set is called vector of consumption, and denoted by $ x = (x_1, x_2, ..., x_n) $ where $ x_i $ with $ i = 1,2, ..., n $, is the quantity consumed of good $i$. Denote by $ X $, the feasible set of these vectors which will be called the set of consumption, usually in economic applications we have $ X \subset \mathbb{R} ^ n_ + $.
In the classical approach of demand theory, the analysis of consumer behavior starts specifying a preference relation over the set $X,$ denoted by $\succeq$. The notation: $ "x \succeq y " $ means that "$ x $ is at least as good as $ y $" or "$ y $ is not preferred to $x$". This preference relation $ \succeq $ is assumed rational, i.e, is complete because the consumer is able to order all possible combinations of goods, and transitive, because consumer preferences are consistent, which means if the consumer prefers $\bar{x}$ to $\bar{y} $ and $\bar{y}$ to $\bar{z}$, then he prefers $\bar{x}$ to $\bar{z} $ (see Definition 3.B.1 of Mas-Colell et al. \cite{Colell}).
The quasiconvex model for a convex preference relation $\succeq,$ is ${\ max}\{ \mu(x) :x \in X\},
$ where $\mu$ is the utility function representing the preference, see Papa Quiroz et al. \cite{PapaLenninOliveira} for more detail. Now consider a multiple criteria, that is, consider $ m $ convex preference relations denoted by $\succeq_i, i=1,2,...,m.$ Suppose that for each preference $\succeq_i,$ there exists an utility function, $ \mu_i,$ respectively, then the problem of maximizing the consumer preference on $ X $ is equivalent to solve the quasiconcave multiobjective optimization problem
\begin{eqnarray*}
\textnormal{(P')\ max}\{ (\mu_{1}(x), \mu_{2}(x), ..., \mu_{m}(x)) \in \mathbb{R}^m :x \in X\}.
\end{eqnarray*}
Since there is not a single point which maximize all the functions simultaneously the concept of optimality is established in terms of Pareto optimality or efficiency. Taking F = $ (- \mu_1, - \mu_2, ..., - \mu_m) $, we obtain a minimization problem with quasiconvex multiobjective function, since each component function is quasiconvex one.
| 937 | 30,721 |
en
|
train
|
0.41.4
|
\section{Exact algorithm}
In this section, to solve the problem $(\ref {pom3}),$ we propose a linear scalarization proximal point algorithm with quadratic regularization using the Fréchet subdifferential, denoted by {\bf SPP} algorithm. \\ \\
{\bf SPP Algorithm }
\begin{description}
\item [\bf Initialization:] Choose an arbitrary starting point
\begin{eqnarray}
x^0\in\mathbb{R}^n
\label{inicio3}
\end{eqnarray}
\item [Main Steps:] Given $x^k$ finding $x^{k+1} $ such that
\begin{eqnarray}
0 \in \hat{\partial}\left( \left\langle F(.), z_k\right\rangle + \dfrac{\alpha_k}{2} \Vert\ .\ - x^k \Vert ^2 + \delta_{\Omega_k}(.) \right) (x^{k+1})
\label{subdiferencial3}
\end{eqnarray}
where $\hat{\partial}$ is the Fréchet subdifferential, $\Omega_k= \left\{ x\in \mathbb{R}^n: F(x) \preceq F(x^k)\right\}$, $\alpha_k > 0 $,\\ $\left\{z_k\right\} \subset \mathbb{R}^m_+\backslash \left\{0\right\}$ and $\left\|z_k\right\| = 1$.
\item [Stop criterion:] If $x^{k+1}=x^{k} $ or $x^{k+1}$ is a Pareto critical point, then stop. Otherwise to do $k \leftarrow k + 1 $ and return to Main Steps.
\end{description}
\subsection{Existence of the iterates}
\begin{teorema}
\label{existe0}
Let $F:\mathbb{R}^n\longrightarrow \mathbb{R}^m\cup \{+ \infty \}^m$ be a vector function satisfying $\bf (C_{1.1}),$ and $\bf (C_{1.2}),$ then the sequence $\left\{x^k\right\}$, generated by the ${\bf SPP}$ algorithm, is well defined.
\end{teorema}
\begin{proof}
Let $x^0 \in \mathbb{R}^n $ be an arbitrary point given in the initialization step. Given $x^k$, define $\varphi_k(x)=\left\langle F(x), z_k\right\rangle + \frac{\alpha_k}{2}\left\|x - x^k\right\|^2 +\delta_{\Omega_k}(x)$, where $\delta_{\Omega_k}(.)$ is the indicator function of ${\Omega_k}$. Then we have that min$\{\varphi_k(x): x \in \mathbb{R}^n\}$ is equivalent to min$\{\left\langle F(x), z_k\right\rangle + \frac{\alpha_k}{2}\left\|x - x^k\right\|^2: x \in \Omega_k\}$. As $\varphi_k$ is lower semicontinuous and coercive then, using Proposition \ref{coercivaesemicont}, we obtain that there exists $x^{k+1} \in \mathbb{R}^n$ which is a global minimum of $\varphi_k.$ From Proposition \ref{otimo}, $x^{k+1}$ satisfies:
$$ 0 \in \hat{ \partial}\left( \left\langle F(.), z_k\right\rangle + \dfrac{\alpha_k}{2} \Vert\ .\ - x^k \Vert ^2 + \delta_{\Omega_k}(.)\right) (x^{k+1})$$
\end{proof}
\subsection{Fejér convergence Property}
To obtain some desirable properties it is necessary to assume the following assumptions on the function $F$ and the initial point $x^0$ :
\begin{description}
\item [$\bf (C_2)$] $F$ is $\mathbb{R}^m_+$-quasiconvex;
\item [${\bf (C_3)}$] The set $\left(F(x^0) - \mathbb{R}^m_+\right)\cap F(\mathbb{R}^n)$ is $\mathbb{R}^m_+$ - complete, meaning that for all sequences $\left\{a_k\right\}\subset\mathbb{R}^n$, with $a_0 = x^0$, such that $F(a_{k+1}) \preceq F(a_k)$, there exists $ a \in \mathbb{R}^n$ such that $F(a)\preceq F(a_k), \ \forall \ k \in \mathbb{N}$.
\end{description}
\begin{obs}
The assumption $ {\bf (C_3)}$ is cited in several works involving the proximal point method for convex functions, see Bonnel et al. \cite{Iusem}, Ceng and Yao \cite {Ceng} and, Villacorta and Oliveira \cite {Villacorta}.
\end{obs}
\begin{proposicao}
Let $F:\mathbb{R}^n\longrightarrow \mathbb{R}^m\cup \{+ \infty \}^m$ be a function that satisfies the assumptions $\bf (C_{1.1})$ and $\bf (C_2)$. If $g \in \hat{\partial}\left( \left\langle F(.), z\right\rangle + \delta_{\Omega} \right)(x)$, with $z \in \mathbb{R}^m_+\backslash \left\{0\right\}$, and $F(y) \preceq F(x)$, with $y \in \Omega$, and $\Omega \subset \mathbb{R}^n$ a closed and convex set, then
$\left\langle g , y - x\right\rangle \leq 0$.
\label{propfejer2}
\end{proposicao}
\begin{proof}
Let $t \in \left( 0, 1\right]$, then from the $\mathbb{R}^m_+$-quasiconvexity of $F$ and the assumption that $F(y) \preceq F(x)$, we have: $F_i(ty + (1-t)x) \leq \textrm{max}\left\{F_i(x), F_i(y)\right\} = F_i(x),\ \forall \ i \in \lbrace 1,...m\rbrace$. It follows that for each $z \in \mathbb{R}^m_+\backslash \left\{0\right\}$, we have
\begin{equation}
\left\langle F(ty + (1-t)x) , z\right\rangle \leq \left\langle F(x) , z\right\rangle.
\label{Fz2}
\end{equation}
As $g \in \hat{\partial}\left( \left\langle F(.), z\right\rangle + \delta_{\Omega} \right)(x)$, we obtain
\begin{equation}
\left\langle F(ty + (1-t)x) , z\right\rangle + \delta_{\Omega}(ty + (1-t)x)\geq \left\langle F(x) , z\right\rangle + \delta_{\Omega}(x) + t\left\langle g, y - x\right\rangle + o(t\left\|y - x\right\|)
\label{soma2}
\end{equation}
From $(\ref{Fz2})$ and $(\ref{soma2})$, we conclude
\begin{equation}
t\left\langle g, y - x\right\rangle + o(t\left\|y - x\right\|) \leq 0
\label{somafim2}
\end{equation}
\\
On the other hand, we have
$\lim\limits_{t \rightarrow 0}\frac{o(t\left\|y - x\right\|)}{t\left\|y - x\right\|}= 0$. Thus,
$\lim\limits_{t\rightarrow 0}\frac{o(t\left\|y - x\right\|)}{t}=\lim\limits_{t\rightarrow 0}\frac{o(t\left\|y - x\right\|)}{t\left\|y - x\right\|}\left\|y - x\right\|=0$.
Therefore, dividing $(\ref{somafim2})$ by $t$ and taking $t \rightarrow 0$, we obtain the desired result.
\end{proof}
Observe that if the sequence $\left\{x^k\right\}$ generated by the {\bf SPP} algorithm satisfies the assumption ${\bf (C_3)}$ then the set
\begin{center}
$E = \left\{x \in \mathbb{R}^n: F\left(x\right)\preceq F\left(x^k\right),\ \ \forall\ k \in \mathbb{N}\right\}$
\end{center}
is nonempty.
\begin{proposicao}
Under assumptions ${\bf(C_{1.1})}$, ${\bf(C_{1.2})},$ ${\bf(C_2)}$ and ${\bf(C_3)}$ the sequence $\left\{x^k\right\}$, generated by the {\bf SPP} algorithm, $(\ref{inicio3})$ and $(\ref{subdiferencial3}),$ is Fejér convergent to $E$.
\label{fejer0}
\end{proposicao}
\begin{proof} Observe that $\forall \ x \in \mathbb{R}^n$:
\begin{eqnarray}
\left\|x^k - x\right\|^2 = \left\|x^k - x^{k+1}\right\|^2 + \left\|x^{k+1} - x\right\|^2 +
2\left\langle x^k - x^{k+1}, x^{k+1} - x\right\rangle.
\label{norma2}
\end{eqnarray}
From Theorem \ref{existe0}, $(\ref{subdiferencial3})$ and from Proposition \ref{somafinita}, $(iv)$, we have that there exists $g_k \in \hat{\partial}\left( \left\langle F(.), z_k\right\rangle + \delta_{\Omega_k}\right)(x^{k+1})$ such that:
\begin{equation}
x^k - x^{k+1} = \dfrac{1}{\alpha_k}g_k
\label{xk}
\end{equation}
Now take $x^* \in E$, then $x^* \in \Omega_k$ for all $k \in \mathbb{N}$. Combining $(\ref{norma2})$ with $x = x^*$ and $(\ref{xk})$, we obtain:
{\footnotesize
\begin{eqnarray}
\left\|x^k - x^*\right\|^2 = \left\|x^k - x^{k+1}\right\|^2 + \left\|x^{k+1} - x^* \right\|^2 +
\frac{2}{\alpha_k}\left\langle g_k ,\ x^{k+1} -x^*\right\rangle
\geq \left\|x^k - x^{k+1}\right\|^2 + \left\|x^{k+1} - x^* \right\|^2\nonumber \\
\label{desigualdade0}
\end{eqnarray}
}
where the last inequality follows from Proposition \ref{propfejer2}. From $(\ref{desigualdade0})$, it implies that
\begin{eqnarray}
0\leq \left\|x^{k+1} - x^k\right\|^2 \leq \left\|x^k - x^*\right\|^2 - \left\|x^{k+1} - x^*\right\|^2.
\label{desigual0}
\end{eqnarray}
Thus,
\begin{equation}
\left\|x^{k+1} - x^*\right\| \leq \left\|x^k - x^*\right\|
\label{fejer70}
\end{equation}
\end{proof}
\begin{proposicao}
Under assumptions ${\bf(C_{1.1})}, {\bf(C_{1.2})},$ ${\bf(C_2)}$ and ${\bf(C_3)},$ the sequence $\left\{x^k\right\}$ generated by the {\bf SPP} algorithm, $(\ref{inicio3})$ and $(\ref{subdiferencial3}),$ satisfies
\begin{center}
$\lim \limits_{k\rightarrow +\infty}\left\|x^{k+1} - x^k\right\| = 0$.
\label{decrescente00}
\end{center}
\end{proposicao}
\begin{proof}
It follows from $(\ref{fejer70})$ that $ \forall\ x^* \in E$, $\left\{\left\|x^k - x^*\right\|\right\}$ is a nonnegative and nonincreasing sequence, and hence is convergent. Thus, the right-hand side of $(\ref{desigual0})$ converges to 0 when $k \rightarrow +\infty$, and the result is obtained.
\end{proof}
| 3,158 | 30,721 |
en
|
train
|
0.41.5
|
\subsection{Convergence Analysis I: non differentiable case }
In this subsection we analyze the convergence of the proposed algorithm when $F$ is a non differentiable vector function.
\begin{proposicao}
Under assumptions ${\bf(C_{1.1})}, {\bf(C_{1.2})},$ ${\bf(C_2)}$ and ${\bf(C_3)},$ the sequence $\left\{x^k\right\}$ generated by the {\bf SPP} algorithm converges to some point of $E$.
\label{acumulacao10}
\end{proposicao}
\begin{proof}
From Proposition $\ref{fejer0}$ and Lemma $\ref{fejerlim1}$, $(i)$, $\left\{x^k\right\}$ is bounded, then there exists a subsequence $\left\{x^{k_j}\right\}$ such that $\lim \limits_{j\rightarrow +\infty}x^{k_j} = \widehat{x}$. Since $\left\langle F(.), z\right\rangle$ is lower semicontinuos function for all $z \in \mathbb{R}^m_+ \backslash \left\{0\right\}$ then
$\left\langle F(\widehat x),z\right\rangle \leq \liminf \limits_{j\rightarrow +\infty}\left\langle F(x^{k_j}) , z\right\rangle $. On the other hand, $x^{k+1} \in \Omega_k$ so $\left\langle F(x^{k+1}) , z\right\rangle \leq \left\langle F(x^{k}) , z\right\rangle $. Furthermore, from assumption ${\bf(C_{1.2})}$ the function $\left\langle F(.), z\right\rangle$ is bounded below for each $z \in \mathbb{R}^m_+\backslash \left\{0\right\},$ then, the sequence $\left\{\left\langle F(x^k),z\right\rangle\right\}$ is nonincreasing and bounded below, hence convergent. Therefore
{\small
\begin{center}
$\left\langle F(\widehat {x}),z\right\rangle \leq \liminf \limits_{j\rightarrow +\infty}\left\langle F(x^{k_j}) , z\right\rangle = \lim \limits_{j\rightarrow +\infty}\left\langle F(x^{k_j}) , z\right\rangle = inf_{k\in \mathbb{N}}\left\{\left\langle F(x^k),z\right\rangle\right\}\leq \left\langle F(x^k),z\right\rangle$.
\end{center}
}
It follows that
$\left\langle F(x^k)-F(\widehat{x}),z\right\rangle \geq 0, \forall \ k \in \mathbb{N}, \forall \ z \in \mathbb{R}^m_+ \backslash \left\{0\right\}$. We conclude that $F(x^k) - F(\widehat{x}) \in \mathbb{R}^m_+$, i.e, $F(\widehat{x})\preceq F(x^k), \forall \ k \in \mathbb{N}$. Therefore $\widehat{x}\in E,$ and by Lemma $\ref{fejerlim1}$, $(ii)$, we get the result.
\end{proof}
\subsubsection{Convergence to a weak Pareto solution}
\begin{teorema}
Let $F:\mathbb{R}^n\longrightarrow \mathbb{R}^m $ be a continuous vector function satisfying the assumptions $\bf (C_{1.2}),$ $\bf (C_2)$ and $\bf (C_3)$. If $ \lim \limits_{k\rightarrow +\infty}\alpha_k= 0$ and the iterations are given in the form
\begin{eqnarray}
x^{k+1}\in \textnormal{arg min} \left\{\left\langle F(x), z_k\right\rangle+\frac{\alpha_k}{2}\left\|x - x^k\right\|^2 : x\in\Omega_k\right\},
\label{recursao0}
\end{eqnarray}
then the sequence $\lbrace x^k\rbrace$ converges to a weak Pareto solution of the problem $(\ref{pom3})$.
\end{teorema}
\begin{proof}
Let\ \ $x^{k+1}\in \textnormal{arg min} \left\{\left\langle F(x), z_k\right\rangle+\frac{\alpha_k}{2}\left\|x - x^k\right\|^2 : x\in\Omega_k\right\}$, this implies that
{\small
\begin{eqnarray}
\left\langle F(x^{k+1}), z_k\right\rangle +\frac{\alpha_k}{2}\left\|x^{k+1}-x^k\right\|^2 \leq \left\langle F(x), z_k\right\rangle +
\frac{\alpha_k}{2}\left\|x -x^k\right\|^2,
\label{des0}
\end{eqnarray}
}
$\forall \ x \in \Omega_k$. Since the sequence $\left\{x^k\right\}$ converges to some point of $E$, then exists $x^* \in E$ such that $\lim \limits_{k\rightarrow +\infty}x^{k}= x^*$. Since that $\left\{z_k\right\}$ is bounded, there exists a subsequence $\left\{z_{k_l}\right\}_{l\in \mathbb{N}}$ such that $\lim \limits_{l\rightarrow +\infty}z_{k_l}=\bar{z}$, with $\bar{z} \in \mathbb{R}^m_+\backslash \left\{0\right\}$. Taking $k=k_l$ in $(\ref{des0})$, we have
{\small
\begin{eqnarray}
\left\langle F(x^{k_l+ 1}), z_{k_l}\right\rangle +\frac{\alpha_{k_l}}{2}\left\|x^{k_l+1}-x^{k_l}\right\|^2 \leq \left\langle F(x), z_{k_l}\right\rangle + \frac{\alpha_{k_l}}{2}\left\|x - x^{k_l}\right\|^2.
\label{desi0}
\end{eqnarray}
}
$\forall \ x \in E.$ As
\begin{center}
$ \frac{\alpha_{k_l}}{2}\left\|x^{k_{l+1}}-x^{k_l}\right\|^2 \rightarrow 0$ and $\frac{\alpha_{k_l}}{2}\left\|x - x^{k_l}\right\|^2 \rightarrow0$ when $l \rightarrow +\infty$
\end{center}
and from the continuity of $F$, taking $l \rightarrow +\infty$ in $(\ref{desi0})$, we obtain
\begin{eqnarray}
\left\langle F(x^*), \overline{z}\right\rangle \leq \left\langle F(x), \overline{z}\right\rangle, \forall \ x \in E
\label{minfi0}
\end{eqnarray}
Thus $x^* \in \textrm{arg min} \left\{\left\langle F(x), \overline{z}\right\rangle: x \in E\right\}$.
Now, $\left\langle F(.), \overline{z}\right\rangle$, with $\bar{z} \in \mathbb{R}^m_+\backslash \left\{0\right\}$ is a strict scalar representation of $F$, so a weak scalar representation, then by Proposition $\ref{inclusao}$ we have that $x^* \in \textrm{arg min}_w \left\{F(x):x \in E \right\}$.\\
We shall prove that $x^* \in \textrm{arg min}_w \left\{F(x):x \in \mathbb{R}^n \right\}$. Suppose by contradiction that $x^* \notin \textrm{arg min}_w \left\{F(x):x \in \mathbb{R}^n \right\}$ then there exists $\widetilde{x} \in \mathbb{R}^n$ such that
\begin{equation}
F(\widetilde{x})\prec F(x^*)
\label{pareto0}
\end{equation}
So for $\bar{z} \in \mathbb{R}^m_+\backslash \left\{0\right\}$ it follows that
\begin{equation}
\left\langle F(\widetilde{x}), \bar{z}\right\rangle < \left\langle F(x^*), \bar{z}\right\rangle
\label{d0}
\end{equation}
Since $x^* \in E$, from $(\ref{pareto0})$ we conclude that $\widetilde{x} \in E$. Therefore from $(\ref{minfi0})$ and $(\ref{d0})$ we obtain a contradiction.
\end{proof}\\
\subsubsection{Convergence to a generalized critical point}
\begin{teorema}
Let $F:\mathbb{R}^n\longrightarrow \mathbb{R}^m $ be a continuous vector function satisfying the assumptions $\bf (C_{1.2}),$ $\bf (C_2)$ and $\bf (C_3)$. If $0 < \alpha_k < \tilde{\alpha}$ then the sequence $\lbrace x^k\rbrace$ generated by the {\bf SPP} algorithm, $(\ref{inicio3})$ and $(\ref{subdiferencial3})$ satisfies
$$\lim \limits_{k\rightarrow +\infty}g^{k} = 0,$$
where $g^k \in \hat{\partial}\left( \left\langle F(.), z_k\right\rangle + \delta_{\Omega_k}\right)(x^{k+1})$.
\label{geral}
\end{teorema}
\begin{proof}
From Theorem $\ref{existe0}$, $(\ref{subdiferencial3})$ and from Proposition $\ref{somafinita}$, $(iv)$, there exists a vector $g_k \in \hat{\partial}\left( \left\langle F(.), z_k\right\rangle + \delta_{\Omega_k}\right)(x^{k+1})$ such that $g^k = \alpha_k(x^k - x^{k+1})$. Since $0 < \alpha_k < \tilde{\alpha}$ then
\begin{equation}
0 \leq \Vert g^k\Vert \leq \tilde{\alpha}\left\|x^k - x^{k+1} \right\|
\label{beta0}
\end{equation}
From Proposition $\ref{decrescente00}$, $\lim \limits_{k\rightarrow +\infty}\left\|x^{k+1} - x^k\right\| = 0$, and from $(\ref{beta0})$ we have $\lim \limits_{k\rightarrow +\infty}g^k = 0$.
\end{proof}
\subsubsection{Finite Convergence to a Pareto Optimal Point}
\noindent
Following the paper of Bento et al, \cite{Bento} subsection 4.3, it is possible to prove the convergence of a special particular case of the proposed algorithm to a Pareto optimal point of the problem (\ref{pom3}). Let $F:\mathbb{R}^n\longrightarrow \mathbb{R}^m\cup \{+ \infty \}^m$ be a proper lower semicontinuous convex function and consider the following particular iteration of (\ref{subdiferencial3}):
\begin{eqnarray}
x^{k+1}= \textnormal{arg min} \left\{\left\langle F(x), z\right\rangle+\frac{\alpha_k}{2}\left\|x - x^k\right\|^2 : x\in\mathbb{R}^n\right\},
\label{recursao0f}
\end{eqnarray}
where $z\in \mathbb{R}^m_+\backslash \left\{0\right\}$ such that $||z||=1.$
\begin{Def}
Consider the set of Pareto optimal points of $(\ref{pom3})$, denoted by $Min(F)$ and let $\bar x\in Min(F)$. We say that $Min(F)$ is $W_{F(\bar x)}$-weak sharp minimum for the problem $(\ref{pom3})$ if there exists a constant $\tau>0$ such that
$$
F(x)-F(\bar x)\notin B(0,\tau d(x,W_{F(\bar x)}))-\mathbb{R}^m_{+},\;\; x\in \mathbb{R}^n\backslash \,W_{F(\bar x)},
$$
where $d(x,Z)=\inf \{d(x,z):z\in Z\}$ and $W_{p}=\{x\in \mathbb{R}^n: F(x)=F(p)\}.$
\end{Def}
\begin{teorema}
Let $F$ be a proper lower semicontinuous convex vector function satisfying the assumptions $\bf (C_{1.2}),$ and $\bf (C_3).$ Assume that $\{x^k\}$ is a sequence generated from the {\bf SPP} algorithm with $x^{k+1}$ being generates from $(\ref{recursao0f})$. Consider also that the set of Pareto optimal points of $(\ref{pom3})$ is nonempty and assume that $Min(F)$ is $W_{F(\bar x)}$-weak sharp minimum for the problem $(\ref{pom3})$ with constant $\tau>0$ for some $\bar x\in Min(F).$ Then the sequence $\{x^k\}$ converges, in a finite number of iterations, to a Pareto optimal point.
\end{teorema}
\begin{proof}
Simmilar to the proof of Theorem 4.3 of Bento et al., \cite{Bento}.
\end{proof}
| 3,290 | 30,721 |
en
|
train
|
0.41.6
|
\subsection{Convergence analysis II: Differentiable Case}
In this subsection we analyze the convergence of the method when $F$ satisfies the following assumption:
\begin{description}
\item [$\bf (C_4)$] $F:\mathbb{R}^n\longrightarrow \mathbb{R}^m$ is a continuously differentiable vector function on $\mathbb{R}^n$.
\end{description}
The next proposition characterizes a quasiconvex differentiable vector functions.
\begin{proposicao}
Let $F :\mathbb{R}^n\longrightarrow \mathbb{R}^m $ be a differentiable function satisfying the assumption ${\bf(C_2)}$, ${\bf(C_3)}.$ If $x \in E,$ then $\left\langle \nabla F_i(x^k) , x - x^k\right\rangle \leq 0$, $\forall \ k \in \mathbb{N}$ and $\forall \ i \in \left\{1,...,m\right\}$.
\label{caracterizacaodif}
\end{proposicao}
\begin{proof}
Since \ $F$ is $\mathbb{R}^m_+$-quasiconvex each $F_i,$ $ i = 1,..., m,$ is quasiconvex. Then the result follows from the classical characterization of the scalar differentiable quasiconvex functions, see
$\textrm{see Mangasarian \cite{Mangasarian}, p.134}$.
\end{proof}
\begin{teorema}
\label{conv3}
Let $F:\mathbb{R}^n\longrightarrow \mathbb{R}^m $ be a function satisfying the assumptions $\bf (C_2)$, $\bf (C_3)$ and $\bf (C_4)$. If $0 < \alpha_k < \tilde{\alpha}$, then the sequence $\lbrace x^k\rbrace$ generated by the {\bf SPP} algorithm, $(\ref{inicio3})$ and $(\ref{subdiferencial3}),$ converges to a Pareto critical point of the problem $(\ref{pom3})$.
\label{teoparetocri}
\end{teorema}
\begin{proof}
In Proposition \ref{acumulacao10} we prove that there exists $\widehat{x} \in E$ such that $\lim \limits_{k\rightarrow +\infty}x^{k}= \widehat{x}$. From Theorem $\ref{existe0}$ and $(\ref{subdiferencial3})$, we have
\begin{equation*}
0 \in \hat{\partial}\left( \left\langle F(.), z_k\right\rangle + \dfrac{\alpha_k}{2}\Vert\ .\ - x^k \Vert ^2 + \delta_{\Omega_k}(.) \right) (x^{k+1})
\end{equation*}
Due to Proposition $\ref{somafinita}$, $(iv)$, we have
\begin{center}
$0 \in \nabla\left(\left\langle F(.) , z_k\right\rangle\right)(x^{k+1})+ \alpha_k \left(x^{k+1} - x^k \right) + \mathcal{N}_{\Omega_k}(x^{k+1})$
\end{center}
where $\mathcal{N}_{\Omega_k}(x^{k+1})$ is the normal cone to $\Omega_k$ at $x^{k+1}\in \Omega_k$.\\
So there exists $\nu_k \in \mathcal{N}_{\Omega_k}(x^{k+1})$ such that:
\begin{equation}
0 = \sum_{i=1}^m \nabla F_i(x^{k+1})(z_k)_i + \alpha_k\left(x^{k+1} - x^k \right) + \nu_k.
\label{otimalidade0}
\end{equation}
Since $\nu_k \in \mathcal{N}_{\Omega_k}(x^{k+1})$ then
\begin{equation}
\left\langle \nu_k \ ,\ x - x^{k+1}\right\rangle \leq\ 0,\ \forall \ x \in \Omega_k.
\label{cone0}
\end{equation}
Take $\bar{x} \in E$. By definition of $E$, $\bar{x} \in \Omega_k$ for all $k \in \mathbb{N}$. Combining $(\ref{cone0})$ with $x = \bar{x}$ and $(\ref{otimalidade0})$, we have
{\small
\begin{eqnarray}
\left\langle \sum_{i=1}^m \nabla F_i(x^{k+1})(z_k)_i , \bar{x} - x^{k+1}\right\rangle + \alpha_k\left\langle x^{k+1} - x^k , \bar{x} - x^{k+1}\right\rangle\geq 0.&
\label{cone00}
\end{eqnarray}
}
Since that $\left\{z_k\right\}$ is bounded, then there exists a subsequence $ \left\{z_{k_j}\right\}_{j \in \mathbb{N}}$ such that $\lim \limits_{j\rightarrow +\infty}z_{k_j}= \bar{z}$ with $\bar{z} \in \mathbb{R}^m_+\backslash \left\{0\right\}$. Thus the inequality in $(\ref{cone00})$ becomes
{\small
\begin{eqnarray}
\left\langle \sum_{i=1}^m \nabla F_i(x^{{k_j}+1})(z_{k_j})_i , \bar{x} - x^{{k_j}+1}\right\rangle + \alpha_{k_j}\left\langle x^{{k_j}+1} - x^{k_j} , \bar{x} - x^{{k_j}+1}\right\rangle\geq 0.&
\label{cone000}
\end{eqnarray}
}
Since $\left\{x^k\right\}$ and $\left\{ \alpha_k\right\}$ are bounded, $\lim \limits_{k\rightarrow +\infty}\left\|x^{k+1} - x^k\right\| = 0$ and $F$ is continuously differentiable, the inequality in $(\ref{cone000})$, for all $\bar{x}\in E$, becomes:
{\small
\begin{eqnarray}
\left\langle\sum_{i=1}^m \nabla F_i(\widehat{x})\bar{z}_i \ ,\ \bar{x} - \widehat{x}\right\rangle \geq 0
\mathbb{R}ightarrow \sum_{i=1}^m \bar{z}_i\left\langle \nabla F_i(\widehat{x}) \ ,\ \bar{x} - \widehat{x} \right\rangle\geq 0.
\label{somai0}
\end{eqnarray}
}
From the quasiconvexity of each component function $F_i$, for each $i \in \left\{1,...,m\right\}$, we have that\\
$\left\langle \nabla F_i(\widehat{x})\ ,\ \bar{x} - \widehat{x} \right\rangle\leq 0$ and because $\bar{z} \in \mathbb{R}^m_+\backslash \left\{0\right\}$, from $(\ref{somai0})$, we obtain
\begin{eqnarray}
\sum_{i=1}^m \bar{z}_i\left\langle \nabla F_i(\widehat{x}) \ ,\ \bar{x} - \widehat{x} \right\rangle = 0.
\label{somai00}
\end{eqnarray}
Without loss of generality consider the set $J = \left\{i \in I: \bar{z}_i > 0 \right\}$, where $I = \left\{1,...,m\right\}$. Thus, from $(\ref{somai00})$, for all $\bar{x} \in E$ we have
\begin{eqnarray}
\left\langle \nabla F_{i}(\widehat{x}) \ ,\ \bar{x} - \widehat{x} \right\rangle = 0,\ \forall \ \ i \in J.
\label{ecritico0}
\end{eqnarray}
Now we will show that $\widehat{x}$ is a Pareto critical point. \\
Suppose by contradiction that $\widehat{x}$ is not a Pareto critical point, then there exists a direction $v \in \mathbb{R}^n$ such that $JF(\widehat{x})v \in -\mathbb{R}^m_{++}$, i.e,
\begin{eqnarray}
\left\langle \nabla F_i(\widehat{x}) , v \right\rangle < 0, \forall \ i \in \left\{1,...,m\right\}.
\label{direcao10}
\end{eqnarray}
Therefore $v$ is a descent direction for the multiobjective function $F$ in $\widehat{x}$, so, $\exists \ \varepsilon > 0$ such that
\begin{eqnarray}
F(\widehat{x} + \lambda v) \prec F(\widehat{x}),\ \forall \ \lambda \in (0, \varepsilon].
\label{descida10}
\end{eqnarray}
Since \ $\widehat{x}\ \in\ E$, then from $(\ref{descida10})$ we conclude that $\widehat{x} + \lambda v \in E$. Thus, from $(\ref{ecritico0})$ with $\bar{x} = \widehat{x} + \lambda v $, we obtain: {\small $\left\langle \nabla F_{i}(\widehat{x})\ ,\ \widehat{x} + \lambda v - \widehat{x} \right\rangle = \left\langle \nabla F_{i}(\widehat{x})\ ,\ \lambda v \right\rangle = \lambda\left\langle \nabla F_{i}(\widehat{x})\ ,\ v \right\rangle = 0$}.\\
It follows that $\left\langle \nabla F_{i}(\widehat{x})\ ,\ v \right\rangle = 0$ for all $ i \in J,$ contradicting $(\ref{direcao10})$. Therefore $\widehat{x}$ is Pareto critical point of the problem $(\ref{pom3})$.
\end{proof}
\section{An inexact proximal algorithm}
In this section we present an inexact version of the {\bf SPP} algorithm, which we denote by {\bf ISPP} algorithm.
| 2,528 | 30,721 |
en
|
train
|
0.41.7
|
\subsection{{\bf ISPP} Algorithm}
Let $F: \mathbb{R}^n \rightarrow \mathbb{R}^m$ be a vector function satisfying the assumptions ${(\bf C_2)}$ and ${(\bf C_4)}$, and consider two sequences: the proximal parameters $\left\{\alpha_k\right\}$ and the sequence $ \left\{z_k \right\}\subset \mathbb{R}^m_+\backslash \left\{0\right\}$ with $\left\|z_k\right\| = 1$.
\begin{description}
\item [\bf Initialization:] Choose an arbitrary starting point
\begin{eqnarray}
x^0\in\mathbb{R}^n
\label{inicio2}
\end{eqnarray}
\item [Main Steps:] Given $x^k,$ define the function $\Psi_k : \mathbb{R}^n \rightarrow \mathbb{R} $ such that $ \Psi_k (x) = \left\langle F(x), z_k\right\rangle $ and consider $\Omega_k = \left\{ x\in \mathbb{R}^n: F(x) \preceq F(x^k)\right\}$. Find $x^{k+1}$ satisfying
\begin{equation}
0 \in \hat{\partial}_{\epsilon_k}\Psi_k (x^{k+1}) + \alpha_k\left(x^{k+1} - x^k\right) + \mathcal{N}_{\Omega_k}(x^{k+1}),
\label{diferencial}
\end{equation}
\begin{equation}
\label{delta}
\displaystyle \sum_{k=1}^{\infty} \delta_k < +\infty,
\end{equation}
where $\delta_k = \textnormal{max}\left\lbrace \dfrac{\varepsilon_k}{\alpha_k}, \dfrac{\Vert \nu_k\Vert}{\alpha_k}\right\rbrace,$ $\varepsilon_k\geq 0,$ and $\hat{\partial}_{\varepsilon_k}$ is the Fréchet $\varepsilon_k$-subdifferential.
\item [Stop criterion:] If $x^{k+1}=x^{k} $ or $x^{k+1}$ is a Pareto critical point, then stop. Otherwise to do $k \leftarrow k + 1 $ and return to Main Steps.
\end{description}
\subsubsection{Existence of the iterates}
\begin{proposicao}
Let $F:\mathbb{R}^n\longrightarrow \mathbb{R}^m $ be a vector function satisfying the assumptions $\bf (C_{1.2}),$ $\bf (C_2)$ and $\bf (C_4)$. Then the sequence $\left\{x^k\right\}$ generated by the {\bf ISPP} algorithm, is well defined.
\label{iteracao2}
\end{proposicao}
\begin{proof}
Consider $x^0 \in \mathbb{R}^n $ given by (\ref{inicio2}). Given $x^k$, we will show that there exists $x^{k+1}$ satisfying the condition $(\ref{diferencial})$. Define the function $\varphi_k(x) = \Psi_k (x)+\frac{\alpha_k}{2}\left\|x - x^k\right\|^2 + \delta_{\Omega_k}(x)$. Analogously to the proof of Theorem\ $\ref{existe0}$ there exists $x^{k+1} \in \Omega_k$ which is a global minimum of $\varphi_k (.),$ so, from Proposition $\ref{otimo}$, $x^{k+1}$ satisfies
\begin{center}
$ 0 \in \hat{ \partial}\left( \Psi_k(.) + \dfrac{\alpha_k}{2}\Vert\ .\ - x^k \Vert ^2 + \delta_{\Omega_k}(.)\right) (x^{k+1})$.
\end{center}
From Proposition $\ref{somafinita},$ $\ (iii)$ and $\ (iv)$, we obtain
\begin{center}
$0 \in \hat{\partial}\Psi_k(x^{k+1})+ \alpha_k \left(x^{k+1} - x^k \right) + \mathcal{N}_{\Omega_k}(x^{k+1})$.
\end{center}
From Remark $\ref{frechet2}$, $x^{k+1}$ satisfies $(\ref{diferencial})$ with $\varepsilon_k = 0$.
\end{proof}
\begin{obs}
\label{bregman}
From the inequality $(a-1/2)^2\geq 0, \forall a\in \mathbb{R},$ we obtain the following relation
\begin{center}
$\Vert x - z\Vert^2 + \frac{1}{4} \geq \Vert x - z\Vert, \ \forall x,z \in \mathbb{R}^n$
\end{center}
\end{obs}
\begin{proposicao}
\label{fejer3}
Let $\left\{x^k\right\}$ be a sequence generated by the {\bf ISPP} algorithm. If the assumptions ${\bf(C_{1.2})},$ ${\bf(C_2)}$, ${\bf(C_3)}$, ${\bf(C_4)}$ and $(\ref{delta})$ are satisfied, then for each $\hat{x} \in E$, $\{ \left\|\hat{x} - x^k\right\|^ 2 \}$ converges and $\{x^k\}$ is bounded.
\end{proposicao}
\begin{proof}
From $(\ref{diferencial})$, there exist $g_k \in \hat{\partial}_{\varepsilon_k}\Psi_k (x^{k+1})$ and $\nu_k \in \mathcal{N}_{\Omega_k}(x^{k+1})$ such that
\begin{center}
$0 = g_k + \alpha_k\left(x^{k+1} - x^k \right) + \nu_k$.
\end{center}
It follows that for any $x \in \mathbb{R}^n$, we obtain
\begin{equation*}
\langle - g_k , x - x^{k+1}\rangle + \alpha_k\langle x^k - x^{k+1}, x - x^{k+1}\rangle = \langle\nu_k, x - x^{k+1}\rangle \leq \Vert \nu_k\Vert \Vert x - x^{k+1}\Vert
\label{diferenca2}
\end{equation*}
Therefore
\begin{equation}
\langle x^k - x^{k+1}, x - x^{k+1} \rangle \leq \dfrac{1}{\alpha_k}\left( \langle g_k , x - x^{k+1}\rangle + \Vert \nu_k\Vert \Vert x - x^{k+1}\Vert\right) .
\label{a}
\end{equation}
Note that $\forall \ x \in \mathbb{R}^n$:
\begin{eqnarray}
\left\|x - x^{k+1}\right\|^2 &-&\left\| x - x^k \right\|^2 \leq \ \ 2\left\langle x^k - x^{k+1}, x - x^{k+1} \right\rangle .
\label{norma5}
\end{eqnarray}
From $(\ref{a})$ and $(\ref{norma5})$, we obtain
\begin{equation}
\left\|x - x^{k+1}\right\|^2 -\left\| x - x^k \right\|^2 \leq \dfrac{2}{\alpha_k}\left( \langle g_k , x - x^{k+1}\rangle + \Vert \nu_k\Vert \Vert x - x^{k+1}\Vert\right) .
\label{b}
\end{equation}
On the other hand, let $\Psi_k (x) = \left\langle F(x), z_k\right\rangle$, where $F: \mathbb{R}^n \rightarrow \mathbb{R}^m$ is continuously differentiable vector function, then $ \Psi_k : \mathbb{R}^n \rightarrow \mathbb{R}$ is continuously differentiable with gradient denoted by $\nabla \Psi_k $. From Proposition $\ref{fdif}$, we have
\begin{equation}
\partial_{\varepsilon_{k}}\Psi_k (x) = \nabla \Psi_k (x) + \varepsilon_{k} B,
\end{equation}
where $B$ is the closed unit ball in $\mathbb{R}^n$ centered at zero. Futhermore, $\hat{\partial}_{\varepsilon_k} \Psi_k(x) \subset \partial_{\varepsilon_k} \Psi_k(x)$, (see (2.12) in Jofré et al. \cite{Jofre}). As $g_k \in \hat{\partial}_{\epsilon_k}\Psi_k(x^{k+1})$, we have that $g_k \in \partial_{\epsilon_k}\Psi_k(x^{k+1})$, then $$g_k = \nabla\Psi_k(x^{k+1}) + \varepsilon_k h_k,$$ with $\Vert h_k \Vert \leq 1 $. Now take $\hat{x}\in \textnormal{E},$ then
\begin{eqnarray}
\langle g_k,\hat{x} - x^{k+1} \rangle & =& \left\langle \nabla \Psi_k(x^{k+1}) + \varepsilon_k h_k\ , \ \hat{x} - x^{k+1}\right\rangle \nonumber \\
&=&\sum_{i=1}^m \left\langle \nabla F_i(x^{k+1})\ ,\ \hat{x} - x^{k+1}\right\rangle (z_k)_i +\varepsilon_k\left\langle h_k\ ,\ \hat{x} -x^{k+1} \right\rangle \nonumber\\
\label{fe1}
\end{eqnarray}
From Proposition $\ref{caracterizacaodif}$, we conclude that $(\ref{fe1})$ becomes
\begin{eqnarray}
\langle g_k,\hat{x} - x^{k+1} \rangle \leq \varepsilon_k\left\langle h_k\ ,\ \hat{x} -x^{k+1} \right\rangle \leq \varepsilon_k \Vert\hat{x} - x^{k+1} \Vert \nonumber\\
\label{f2}
\end{eqnarray}
From Remark $\ref{bregman}$ with $x = \hat{x}$ and $z = x^{k+1}$, follows
\begin{eqnarray}
\Vert \hat{x} - x^{k+1}\Vert \leq \left( \Vert \hat{x} - x^{k+1}\Vert ^2 + \frac{1}{4}\right).
\label{c}
\end{eqnarray}
Consider $x = \hat{x}$ in $(\ref{b})$, using $(\ref{f2})$, $(\ref{c})$ and the condition (\ref{delta}) we obtain
\begin{eqnarray*}
\left\|\hat{x} - x^{k+1}\right\|^2 -\left\| \hat{x} - x^k \right\|^2 & \leq & \dfrac{2}{\alpha_k}\left( \varepsilon_k + \Vert \nu_k\Vert\right)\Vert \hat{x} - x^{k+1}\Vert \\
&\leq & 4 \delta_k \left\|\hat{x} - x^{k+1}\right\|^2 + \delta_k.
\end{eqnarray*}
Thus
\begin{eqnarray}
\left\|\hat{x} - x^{k+1}\right\|^2 \leq \left(\frac{1}{1-4\delta_k}\right)\left\|\hat{x} - x^{k}\right\|^2+ \frac{\delta_k}{1-4\delta_k}.
\label{d}
\end{eqnarray}
The condition (\ref{delta}) guarantees that
\begin{center}
$\delta_k < \dfrac{1}{4}, \ \ \forall k > k_0, $\\
\end{center}
where $k_0$ is a natural number sufficiently large, and so,
\begin{equation*}
1 \leq \frac{1}{1-4\delta_k} \leq 1+2\delta_k <2,\ \ \ \textnormal{for}\ \ k \geq k_0,
\end{equation*}
combining with $(\ref{d})$, results in
\begin{equation}
\left\|\hat{x} - x^{k+1}\right\|^2 \leq \left(1 + 2\delta_k \right)\left\|\hat{x} - x^{k}\right\|^2 + 2\delta_k .
\label{e2}
\end{equation}
Since $\displaystyle \sum_{i=1}^{\infty} \delta_k < \infty,$ applying Lemma $\ref{p}$ in the inequality $(\ref{e2})$, we obtain the convergence of $\{ \|\hat{x} - x^k\|^2 \}$, for each $\hat{x} \in E,$ which implies that there exists $M \in \mathbb{R}_+$, such that $\left\|\hat{x} - x^k \right\| \leq M, \ \ \forall \ k \in \mathbb{N}.
$
Now, since that $\Vert x ^k\Vert \leq \Vert x ^k - \hat{x} \Vert + \Vert \hat{x} \Vert,$ we conclude that $\{x^k\}$ is bounded, and so, we guarantee that the set of accumulation points of this sequence is nonempty. \end{proof}
| 3,290 | 30,721 |
en
|
train
|
0.41.8
|
\subsubsection{Convergence of the {\bf ISPP} algorithm}
\begin{proposicao}{(\bf Convergence to some point of E)}\\
If the assumptions ${\bf(C_{1.2})},$ ${\bf(C_2)}$, ${\bf(C_3)}$ and ${\bf(C_4)}$ are satisfied, then the sequence $\left\{x^k\right\}$ generated by the {\bf ISPP} algorithm converges to some point of the set $E$.
\label{acumulacao100}
\end{proposicao}
\begin{proof}
As $\left\{x^k\right\}$ is bounded, then there exists a subsequence $\left\{x^{k_j}\right\}$ such that $\lim \limits_{j\rightarrow +\infty}x^{k_j} = \widehat{x}$. Since $F$ is continuous in $\mathbb{R}^n$, then the function $\left\langle F(.), z\right\rangle$ is also continuous in $\mathbb{R}^n$ for all $z \in \mathbb{R}^m$, in particular, for all $z \in \mathbb{R}^m_+ \backslash \left\{0\right\}$, and
$\left\langle F(\widehat x),z\right\rangle = \lim \limits_{j\rightarrow +\infty}\left\langle F(x^{k_j}) , z\right\rangle $. On the other hand, we have that $F(x^{k+1}) \preceq F(x^{k})$, and so, $\left\langle F(x^{k+1}) , z\right\rangle \leq \left\langle F(x^{k}) , z\right\rangle $ for all $z \in \mathbb{R}^m_+ \backslash \left\{0\right\}$. Furthermore the function $\left\langle F(.), z\right\rangle$ is
bounded below, for each $z \in \mathbb{R}^m_+\backslash \left\{0\right\}$, then the sequence $\left\{\left\langle F(x^k),z\right\rangle\right\}$ is nonincreasing and bounded below, thus convergent. So,
{\small
\begin{center}
$\left\langle F(\widehat {x}),z\right\rangle = \lim \limits_{j\rightarrow +\infty}\left\langle F(x^{k_j}) , z\right\rangle = \lim \limits_{j\rightarrow +\infty}\left\langle F(x^{k}) , z\right\rangle = inf_{k\in \mathbb{N}}\left\{\left\langle F(x^k),z\right\rangle\right\}\leq \left\langle F(x^k),z\right\rangle$.
\end{center}
}
It follows that $F(x^k) - F(\widehat{x}) \in \mathbb{R}^m_+$, i.e, $F(\widehat{x})\preceq F(x^k), \forall \ k \in \mathbb{N}$. Therefore $\widehat{x}\in E$. Now, from Proposition $\ref{fejer3}$, we have that the sequence $\{ \left\|\widehat{x} - x^k\right\|\}$ is convergent, and since $\lim \limits_{k\rightarrow +\infty}\left\|x^{k_j} - \hat{x}\right\| = 0$, we conclude that $\lim \limits_{k\rightarrow +\infty}\left\|x^{k} - \widehat{x}\right\| = 0$, i.e, $\lim \limits_{k\rightarrow +\infty}x^{k} = \widehat{x}.$
\end{proof}
\begin{teorema}
\label{conv4}
Suppose that the assumptions ${\bf(C_{1.2})},$ ${\bf(C_2)}$, ${\bf(C_3)}$ and ${\bf(C_4)}$ are satisfied. If $0 < \alpha_k < \widetilde{\alpha}$, then the sequence $\lbrace x^k\rbrace$ generated by the {\bf ISPP} algorithm , $(\ref{inicio2})$, $(\ref{diferencial})$ and $(\ref{delta}),$ converges to a Pareto critical point of the problem $(\ref{pom3})$.
\end{teorema}
\begin{proof}
From Proposition $\ref{acumulacao100}$ there exists $\widehat{x}\in E$ such that $\lim \limits_{j\rightarrow +\infty}x^{k}= \widehat{x}$. Furthermore, as the sequence $\left\{z^k\right\}$ is bounded, then there exists $ \left\{z^{k_j}\right\}_{j \in \mathbb{N}}$ such that $\lim \limits_{j\rightarrow +\infty}z^{k_j}= \bar{z}$, with $\bar{z} \in \mathbb{R}^m_+\backslash \left\{0\right\}$. From $(\ref{diferencial})$ there exists $g_{k_j} \in \hat{\partial}_{\varepsilon_{k_j}}\Psi_{k_j} (x^{{k_j}+1})$, with $g_{k_j} = \nabla\Psi_{k_j}(x^{{k_j}+1}) + \varepsilon_{k_j} h_{k_j}$ with $\Vert h_{k_j} \Vert \leq 1 $, and $\nu_{k_j} \in \mathcal{N}_{\Omega_{k_j}}(x^{{k_j}+1})$, such that:
\begin{equation}
0 = \sum_{i=1}^m \nabla F_i(x^{{k_j}+1})(z_{k_j})_i + \varepsilon_{k_j} h_{k_j} + \alpha_{k_j} \left(x^{{k_j}+1} - x^{k_j} \right) + \nu_{k_j}
\label{otimalidade2}
\end{equation}
Since $\nu_{k_j} \in \mathcal{N}_{\Omega_{k_j}}(x^{{k_j}+1})$ then,
\begin{equation}
\left\langle \nu_{k_j} \ ,\ x - x^{{k_j}+1}\right\rangle \leq\ 0,\ \forall \ x \in \Omega_{k_j}
\label{cone6}
\end{equation}
Take $\bar{x} \in E$. By definition of $E$, $\bar{x} \in \Omega_k$, for all $ k \in \mathbb{N}$, so $\bar{x} \in \Omega_{k_j}$. Combining $(\ref{cone6})$ with $x = \bar{x}$ and $(\ref{otimalidade2})$, we have
{\footnotesize
\begin{eqnarray}
0 &\leq & \left\langle \sum_{i=1}^m \nabla F_i(x^{{k_j}+1})(z_{k_j})_i\ ,\ \bar{x} - x^{{k_j}+1}\right\rangle + \varepsilon_{k_j}\left\langle h_{k_j}\ ,\ \bar{x} - x^{{k_j}+1}\right\rangle +
+\alpha_{k_j}\left\langle x^{{k_j}+1} - x^{k_j}\ ,\ \bar{x} - x^{{k_j}+1}\right\rangle \nonumber \\
&\leq & \left\langle\sum_{i=1}^m \nabla F_i(x^{{k_j}+1})(z_{k_j})_i\ ,\ \bar{x} - x^{{k_j}+1}\right\rangle + \varepsilon_{k_j}M + \tilde{\alpha}\left\langle x^{{k_j}+1} - x^{k_j}\ ,\ \bar{x} - x^{{k_j}+1}\right\rangle \nonumber \\
\label{aaa}
\end{eqnarray}
}Observe that, $\forall \ x \in \mathbb{R}^n$:
\begin{eqnarray}
\left\|x^{k+1} - x^k \right\|^2 &=& \left\| x - x^k \right\|^2 - \left\| x - x^{k+1}\right\|^2 + 2\left\langle x^k - x^{k+1} , x - x^{k+1} \right\rangle
\label{consecutiva}
\end{eqnarray}
Now, from $(\ref{a})$ with $x = \bar{x} \in E$, and $(\ref{f2})$, we obtain
\begin{eqnarray*}
\left\langle x^k - x^{k+1} ,\bar{x} - x^{k+1} \right\rangle\leq \left\| \bar{x} - x^{k+1}\right\|\left( \dfrac{\varepsilon_k}{\alpha_k} + \dfrac{\Vert \nu_k \Vert}{\alpha_k}\right) \leq 2M\delta_k
\end{eqnarray*}
Thus, from $(\ref{consecutiva})$, with $x = \bar{x}$, we have
\begin{eqnarray}
0 \leq \left\|x^{k+1} - x^k \right\|^2 \leq \left\| \bar{x} - x^k \right\|^2 - \left\| \bar{x} - x^{k+1}\right\|^2 +4M\delta_k
\label{consecutiva2}
\end{eqnarray}
Since that the sequence $\left\{ \left\|\bar{x} - x^k\right\|\right\}$ is convergent and $\displaystyle\sum_{i=1}^{\infty}\delta_k < \infty$, from $(\ref{consecutiva2})$ we conclude that $\displaystyle \lim_{k \to +\infty} \left\|x^{k +1} - x^k \right\| = 0$. Furthermore, as
\begin{eqnarray}
0 \leq \left\|x^{{k_j}+1} - \bar{x} \right\| \leq \Vert x^{{k_j}+1} - x^{k_j} \Vert + \Vert x^{k_j} - \bar{x}\Vert,
\label{zero}
\end{eqnarray}
we obtain that the sequence $\{ \left\|\bar{x} - x^{{k_j}+1}\right\|\}$ is bounded.\\
Thus returning to $(\ref{aaa})$, since $\lim \limits_{k\rightarrow+\infty}\varepsilon_k = 0 $, $\lim \limits_{j\rightarrow +\infty}x^{k}= \widehat{x}$ and $\lim \limits_{j\rightarrow +\infty}z^{k_j}= \bar{z}$, taking $j \rightarrow + \infty$, we obtain
\begin{equation}
\sum_{i=1}^m \bar{z}_i\left\langle \nabla F_i(\widehat{x}) \ ,\ \bar{x} - \widehat{x} \right\rangle\geq 0.
\label{cc}
\end{equation}
Therefore, analogously to the proof of Theorem $\ref{teoparetocri}$, starting in $(\ref{somai0})$, we conclude that $\widehat{x}$ is a Pareto critical point to the problem $(\ref{pom3})$.
\end{proof}
| 2,675 | 30,721 |
en
|
train
|
0.41.9
|
\section{Finite convergence to a Pareto optimal point}
\noindent
In this section we prove the finite convergence of a particular inexact scalarization proximal point algorithm for proper lower semicontinuous convex functions, which we call Convex Inexact Scalarization Proximal Point algorithm, {\bf CISPP} algorithm.
Let $F:\mathbb{R}^n\longrightarrow \mathbb{R}^m\cup \{+ \infty \}^m$ be a proper lower semicontinuous convex function and consider $ z \in \mathbb{R}^m_+\backslash \left\{0\right\}$ with $\left\|z\right\| = 1$ and the sequences of the proximal parameters $\left\{\alpha_k\right\}$ such that $0<\alpha_k<\bar{\alpha}.$
\begin{description}
\item [{\bf CISPP algorithm}]
\item [\bf Initialization:] Choose an arbitrary starting point
\begin{eqnarray}
x^0\in\mathbb{R}^n
\label{inicio2c}
\end{eqnarray}
\item [Main Steps:] Given $x^k,$ and find $x^{k+1}$ satisfying
\begin{equation}
e^k\in \partial \left( \langle F(.), z\rangle+ \frac{\alpha_k}{2}\|. - x^k\|^2 \right)(x^{k+1})
\label{recursao0f2c}
\end{equation}
\begin{equation}
\label{deltac}
\displaystyle \sum_{k=1}^{\infty} ||e^k||<+\infty,
\end{equation}
where $\partial$ is the classical subdifferential for convex functions.
\item [Stop criterion:] If $x^{k+1}=x^{k} $ or $x^{k+1}$ is a Pareto optimal point, then stop. Otherwise to do $k \leftarrow k + 1 $ and return to Main Steps.
\end{description}
\begin{teorema}
Let $F:\mathbb{R}^n\longrightarrow \mathbb{R}^m\cup \{+ \infty \}^m$ be a proper lower semicontinuous convex function, $0\preceq F$ and assume that $\{x^k\}$ is a sequence generated by the {\bf CISPP} algorithm, $(\ref{inicio2c})$, $(\ref{recursao0f2c})$ and $(\ref{deltac})$. Consider also that the set of Pareto optimal points of $(\ref{pom3}),$ denoted by $Min(F),$ is nonempty and assume that $Min(F)$ is $W_{F(\bar x)}$-weak sharp minimum for the problem $(\ref{pom3})$ with constant $\tau>0$ for some $\bar x\in Min(F).$ Then the sequence $\{x^k\}$ converges, in a finite number of iterations, to a Pareto optimal point.
\end{teorema}
\begin{proof}
Denote by $g(x)=\langle F(.), z\rangle$ and $$U=\textnormal{arg min}\{g(x): x\in \mathbb{R}^n\}.$$
As $Min(F)$ is nonempty and it is $W_{F(\bar{x})}$-weak sharp minimum then $Min(F)=WMin(F),$ where $WMin (F)$ denotes the weak Pareto solution of the problem (\ref{pom3}). From Theorem 4.2 of \cite{Bento} it follows that $U$ is nonempty.\\
On the other hand, it is well known that the above {\bf CISPP} algorithm is well defined and converges to some point of $U,$ see
Rockafellar \cite{Rocka}. We will prove that this convergence is obtained to Pareto optimal point in a finite number of iterations.\\
Suppose, by contradiction, that the sequence $\{x^k\}$ is infinite and take $x^*\in U.$
From the iteration $(\ref{recursao0f2c})$ we have that
$$
g(x^{k+1})-g(x^*)\leq \frac{\alpha_k}{2} \left( ||x^k-x^*||^2-||x^{k+1}-x^k||^2\right)+||e^k|||x^{k+1}-x^*||
$$
From (\ref{norma2}) the above inequality implies
$$
g(x^{k+1})-g(x^*)\leq \frac{\alpha_k}{2} \left( ||x^{k+1}-x^*||^2 + 2||x^{k+1}-x^k||||x^{k+1}-x^*||\right)+||e^k|||x^{k+1}-x^*||
$$
Taking $x^*\in U$ such that $||x^{k+1}-x^*||=d(x^{k+1}, W_{F(\bar x)})$ and using the condition of $0<\alpha_k<\bar{\alpha},$ we obtain
$$
\frac{2\tau}{\bar{\alpha}}\leq d(x^{k+1},W_{F(\bar x)})+2||x^{k+1}-x^k|| +\frac{2}{\bar{\alpha}}||e^k||
$$
Letting $k$ goes to infinite in the above inequality we obtain that
$$
\frac{2\tau}{\bar{\alpha}}\leq 0,
$$
which is a contradiction. Thus the {\bf CISPP} algorithm converges to a some point $\hat{x}\in U$ in a finite number of steps.\\
Finally, we will prove that the point of convergence of $\{x^k\},$ denoted by $\hat{x}\in U,$ is a Pareto optimal point of the problem (\ref{pom3}). In fact, as $g$ is weak scalar of the vector function $F,$ then from Proposition \ref{inclusao}, we have $\hat{x}\in WMin(F)$ anf from the equality $Min(F)=WMin(F),$ we obtain that $\hat{x}\in U,$ is a Pareto optimal point of the problem.
\end{proof}
\section{A Numerical Result}
\noindent
In this subsection we give a simple numerical example showing the functionally of the proposed method. For that we use a Intel Core i5 computer 2.30 GHz, 3GB of RAM, Windows 7 as operational system with SP1 64 bits and we implement our code using MATLAB software 7.10 (R2010a).
\begin{exem}
Consider the following multiobjective minimization problem
$$
\min \left\{(F_1(x_1,x_2),F_2(x_1,x_2)): (x_1,x_2)\in \mathbb{R}^2 \right\}
$$
where $F_1(x_1,x_2)=-e^{-x_1^2-x_2^2}+1$ and $F_2(x_1,x_2)=(x_1-1)^2+(x_2-2)^2.$ This problem satisfies the assumptions $\bf (C_{1.2}),$ $\bf (C_2)$ and $\bf (C_4).$ We can easily verify that the points $\bar x=(0,0)$ and $\hat{x}=(1,2)$ are Pareto solutions of the problem.\\
We take $x^0=(-1,3)$ as an initial point and given $x^k\in \mathbb{R}^2,$ the main step of the {\bf SPP} $\mbox{algorithm}$ is to find a critical point ( local minimum, local maximum or a saddle point) of the following problem
$$
\label{e}
\left\{\begin{array}{l}
\min g(x_1,x_2)=(-e^{-x_1^2-x_2^2}+1)z_1^k + \left((x_1-1)^2+(x_2-2)^2 \right)z_2^k+\frac{\alpha_k}{2}\left((x_1-x_1^k)^2+(x_2-x_2^k)^2\right)
\\
s.to:\\
\hspace{0.5cm} x_1^2+x_2^2\leq (x_{1}^k)^2+(x_{2}^k)^2\\
\hspace{0.5cm} (x_1-1)^2+(x_2-2)^2\leq (x_1^k-1)^2+(x_2^k-2)^2
\end{array}\right.
$$
In this example we consider $z_k=\left(z_1^k,z_2^k\right)=\left(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\right)$ and $\alpha_k=1,$ for each $k.$
We take $z^0=(2,3)$ as the initial point to solve all the subproblems using the MATLAB function fmincon (with interior point algorithm) and we consider the stop criterion $||x^{k+1}-x^k||<0.0001$ to finish the algorithm. The numerical results are given in the following table:
{\scriptsize $$ \begin{tabular}{|c|c|c|c|c|c|c|c|}\hline
$k$& $ N[x^{k}] $& $x^k=(x^{k}_{1}, x^{k}_{2}) $ &$ ||x^{k}-x^{k-1} || $& $\sum F_i(x^k)z_i^k$& $ F_1(x_1^k,x_2^k)$ & $F_2(x_1^k,x_2^k)$ \\ \hline
1 & 10 & (0.17128, 2.41010)& 1.31144& 1.30959& 0.99709 &0.85496 \\
2 & 10 & (0.65440, 2.16217) &0.54302 & 0.80586 & 0.99392 & 0.14574 \\
3 & 9 & (0.85337, 2.05877 ) &0.22423 & 0.71983 & 0.99303 & 0.02496 \\
4 & 7 & (0.93534, 2.01588 ) & 0.09251 & 0.70518 & 0.99284 & 0.00443 \\
5 & 7 & (0.96912, 1.99814) & 0.03816 & 0.70268 & 0.99279 & 0.00096 \\
6 & 7 & (0.98305, 1.99080) & 0.01574 & 0.70226 & 0.99277 & 0.00037 \\
7 & 7 & (0.98879, 1.98776) & 0.00649 & 0.70219 & 0.99277 & 0.00028 \\
8 & 7 & (0.99115,1.98651) & 0.00268 & 0.70217 & 0.99276 & 0.00026 \\
9 & 7 & (0.99213, 1.98599) & 0.00110 & 0.70217& 0.99276 & 0.00026 \\
10 & 7 & (0.99253, 1.98578) & 0.00046 & 0.70217 & 0.99276 & 0.00026 \\
11 & 7 & (0.99270, 1.98569) & 0.00019 & 0.70217 & 0.99276 & 0.00026 \\
12 & 7 & (0.99277,1.98565) & 0.00008 & 0.70217 & 0.99276 & 0.00026 \\ \hline
\end{tabular}
$$}
The above table show that we need $k=12$ iterations to solve the problem, $N[x^{k}]$ denotes the inner iterations of each subproblem to obtain the point $x^{k},$ for example to obtain the point $x^3=(0.85337, 2.05877 )$ we need $N[x^3]=9$ inner iterations. Observe also that in each iteration we obtain $F(x^k)\succeq F(x^{k+1})$ and the function $\langle F(x^k),z^k\rangle$ is non increasing.
\end{exem}
\section{Conclusion}
\noindent
This paper introduce an exact linear scalarization proximal point algorithm, denoted by {\bf SPP} algorithm, to solve arbitrary extended multiobjective quasiconvex minimization problems. In the differentiable case it is presented an inexact version of the proposed algorithm and for the (not necessary differentiable) convex case, we present an inexact algorithm and we introduced some conditions to obtain finite convergence to a Pareto optimal point.
To reduce considerably the computational cost in each iteration of the {\bf SPP} algorithm it is need to consider the unconstrained iteration
\begin{equation}
\label{subdiferencialintF}
0 \in \hat{\partial}\left( \left\langle F(.), z_k\right\rangle + \dfrac{\alpha_k}{2} \Vert\ .\ - x^k \Vert ^2 \right) (x^{k+1})
\end{equation}
which is more practical than (\ref{subdiferencial3}). One natural condition to obtain (\ref{subdiferencialintF}) is that $x ^{k +1} \in (\Omega_k)^0$ (interior of $\Omega_k$). So we believe that a variant of the {\bf SPP} algorithm may be an interior variable metric proximal point method.
A future research may be the extension of the proposed algorithm for more general constrained vector minimization problems using proximal distances. Another future research may be to obtain a finite convergence of the {\bf SPP} algorithm for the quasiconvex case.
{\footnotesize
}
\end{document}
| 3,678 | 30,721 |
en
|
train
|
0.42.0
|
\begin{document}
\title{Axiomatizing rectangular grids with no extra non-unary relations}
\begin{abstract}
We construct a formula $\phi$ which axiomatizes non-narrow rectangular grids
without using any binary relations other than the grid neighborship relations.
As a corollary, we prove that a set $A \subseteq {\mathbb N}$ is a spectrum of a formula
which has only planar models if numbers $n \in A$ can be recognized by a
non-deterministic Turing
machine (or a one-dimensional cellular automaton)
in time $t(n)$ and space $s(n)$, where $t(n)s(n) \leq n$ and $t(n),s(n) = \Omega(\log(n))$.
\end{abstract}
\section{Introduction}
The \emph{spectrum} of a $\phi$, denoted ${\rm{spec}}(\phi)$ is the set of cardinalities of
models of $\phi$. Let ${\rm{SPEC}}$ be the set of $A \subseteq {\mathbb N}$ such that $A$ is
a spectrum of some formula $\phi$ is an interesting research area \cite{scholz,fiftyyears};
it is known that SPEC=NE, i.e., $A$ is a spectrum of a first order formula iff the set
of binary representations of the elements of $A$ is in the complexity class NE
\cite{fagin74,JonesS74}.
However, the characterization of spectra remains open if we require our formula, or
our models, to have additional properties. In \cite{planarspectra} we study the
complexity class ${\rm{FPSPEC}}$ (Forced Planar Spectra), which is the set of
$A \subseteq {\mathbb N}$ such that there exists a formula $\phi$ such that
${\rm{spec}}(\phi)=S$ and all models of $\phi$ are planar. It is shown there that
${\rm{FPSPEC}} \supseteq {\rm{NTISP}}(n^{1-\epsilon}, \log(n))$, where ${\rm{NTISP}}(t(n), s(n))$
the set fo $A \subseteq {\mathbb N}$ such that there exists a non-deterministic
Turing machine which
recognizes the binary representation of $n$ in time $t(n)$ and space $s(n)$.
However, this result is not satisfying, since space $\log(n)$ is very low;
a construction of which allows more space is left as an open problem.
In this paper we construct a formula $\phi$ over a signature consisting of only binary relations
$U,D,L,R$ (neighbors in the grid in all directions) and unary relations, and which axiomatizes rectangular grids which
are not {\it narrow}, i.e., grids of dimensions $x^* \times y^*$ where $x^* = \Omega(\log(y^*))$ and $y^* = \Omega(\log(x^*))$.
We show that it is impossible to give a similar axiomatization of rectangular grids which includes the narrow ones.
Non-narrow rectangular grids are planar graphs of bounded degree, and they can be used to simulate Turing machines, and thus we obtain the
following corollary: ${\rm{FPSPEC}} \supseteq {\rm{NTISP}}(t(n), s(n))$ for every
pair of functions $t(n), s(n)$ such that $t(n) \cdot s(n) \leq n$ and $t(n),s(n) = \Omega(\log(n))$.
In fact, we
get a bit more -- we can actually simulate a non-deterministic one-dimensional cellular automaton (1DCA)
working in the given time and memory. While 1DCAs are less commonly taught than Turing machines,
they are simpler to define and more powerful, since they can perform computations on the whole tape at once \cite{ccotb}.
\section{Axiomatizing a rectangular grid}\label{gridax}
We obtain our goal by showing a first-order formula whose all finite models are rectangular grids.
A {\bf rectangular grid} is a relational structure $G=(V(G),L,R,U,D)$ such that $V(G) = \{0..x^*\} \times \{0..y^*\}$, and
the relations $L$, $R$, $U$, $D$ hold only in the following situations:
$L((x,y), (x-1,y))$, $R((x,y), (x+1),y)$, $U((x,y), (x,y-1))$, $D((x,y), (x,y+1))$, as long as these vertices exist.
\paragraph{Geometry}
We will use four binary relations $L$, $R$, $U$, $D$, which correspond to Left, Right, Up, Down,
respectively. We will need axioms to specify that these four relations work according to
the Euclidean square grid geometry.
\begin{itemize}
\item {\bf Partial injectivity.}
Our relations $X \in \{L, R, U, D\}$ are partial injective functions.
That is, we have an axiom $\forall x \forall y X(x,y) \wedge X(x,z) \Rightarrow y=z$. For
$X \in \{L, R, U, D\}$, we will write $X(x)$ for the element $y$ such that $X(x,y)$
(if it exists).
\item {\bf Inverses.} $\forall x \forall y R(x,y) \iff L(y,x) \wedge U(x,y) \iff D(y,x)$. This axiom
formalizes our interpretation of directions (that Left is inverse to Right and Up is inverse to Down).
\item {\bf Commutativity.} Let $H \in \{L,R\}$ and $V \in \{U,D\}$. Then
$\forall x \forall y \forall z H(x,y) \wedge V(x,z) \Rightarrow \exists t H(z,t) \wedge H(y,t)$.
This axiom axiomatizes the Euclidean geometry of our grid: horizontal and vertical movements commute.
Additionally, it enforces that whenever we can go horizontally and vertically from the given $x$,
we can also combine these two movements and move diagonally.
\end{itemize}
\paragraph{Binary Counters}
We will require our grid to know its number of rows. To this end, we will introduce an extra
relation $B_V$. Intuitively, replace every vertex $v$ in the row $r=(x, R(x), R^2(x), \ldots)$,
where $L(x)$ is not defined, with
1 if $B_V(v)$, and 0 otherwise. The axioms in this section will enforce that the obtained number
(written in the little endian binary notation) is the index of our row.
\begin{itemize}
\item {\bf Horizontal Zero.} $\forall x (\neg \exists y U(x,y)) \Rightarrow (\neg B_V(v))$. The
binary number encoded in the first row is zero.
\item {\bf Horizontal Increment.}
To increment a (little endian) binary number, we change every bit which is either the leftmost one,
or such that its left neighbor changed from 1 to 0. This can be written as the following formula:
$\forall x (\exists y U(x,y)) \Rightarrow ((B_v(x) \not \iff B_v(U(x))) \iff C(x)$
where $C(x) = ((\neg\exists y L(x,y)) \vee (\neg B_v(L(x)) \wedge B_v(U(L(x))))$.
\item {\bf No Horizontal Overflow.} $\forall x (\neg \exists y R(x,y)) \Rightarrow (\neg B_V(v))$. This axiom
makes sure that our binary counter does not overflow.
\end{itemize}
We also have analogous axioms for vertical binary counters, using an extra unary relation $B_H$, counting
from right to left, with the least significant bit on the bottom. See Figures \ref{fig}a and \ref{fig}c, where
the vertices of the grid satisfying respectively $B_V$ and $B_H$ are shown (ignore the small white circles and
dark grey boxes for now -- they will be essential for our further construction).
Let $\phi_1$ be the conjunction of all axioms above.
\begin{theorem}\label{tgeo}
If $G$ is a connected finite model of $\phi_1$ and there exists an $v \in V(G)$ and a relation $X \in \{L,R,U,D\}$ such that
$X(v)$ is not defined, then $G$ is a rectangular grid.
\end{theorem}
\begin{proof}
Take $X$ and $v$ such that $X(v)$ be not defined. Without loss of generality we can assume that $X \in \{L,R\}$ (horizontal and
vertical axioms are symmetrical). Furthermore, we can also assume that $X = L$ (since $R$ is the
inverse of $L$, if $R$ is not defined for some element, then so is $L$).
Let $v+(0,y) = D^y(v)$, where $x \geq 0$ and $y \geq 0$. From the commutativity axiom, $L(v+(0,y))$ is not defined for any
$y$. Indeed, if $L(v+(0,y))$ was defined for $y>0$, we have $L(v+(0,y))$ and $U(v+(0,y)) = v+(0,y-1)$ defined, hence
$L(v+(0,y-1))$ is defined too.
Let $b_y = \sum 2^x [B_V(R^x(v+(0,y)))]$. From the Horizontal Increment and No Horizontal Overflow axioms, it is easy to show that $b_{y+1} = b_y+1$.
Furthermore, we have that $b_y < 2^{|V|}$. Therefore, there must exist $y$ such that $D(v+(0,y))$ is not defined. Let $v' = v+(0,y)$.
Let $x^*$ be the greatest x such that $R^x(v')$ is defined, and $y$ be the greatest $y^*$ such that $U^y(v')$ is defined. Let $G = \{0,\ldots,x^*\} \times \{0,\ldots,y^*\}$,
and for $(x,y) \in G$, let $m(x,y) = R^x(U^y(v'))$. It is straightforward that $m$ gives an isomorphism between the rectangular grid $G$ and $V$.
$\rule{2mm}{2mm}$\par
\end{proof}
| 2,636 | 7,338 |
en
|
train
|
0.42.1
|
\section{Forbidding Tori}
However, rectangular grids are not the only models of $\phi_1$. Consider the torus $T = \{0, \ldots, x^*\} \times \{0, \ldots, y^*\}$, where
$(x^*,y)$ is additionally connected (with the $R$ relation) to $(0,y$), and $(x,y^*)$ is additionally connected to $(x,0)$ (with the $U$ relation),
and we add the respective inverses to $L$ and $D$. If $B_V$ and $B_H$ are empty relations, the torus $T$ satisifes all of our axioms. Additionally,
if $G$ is a model of $\phi_1$, then the disjoint union $G \cup T$ is also a model of $\phi_1$.
To prevent this, we use the following result of Berger \cite{berger}.
\begin{theorem}\label{wangth}
There exists a finite set of Wang tiles $K = \{k_1, \ldots, k_t\}$ and relations $T_R, T_D \subseteq K \times K$ such that there exists a tiling
$C: {\mathbb Z} \times {\mathbb Z} \rightarrow K$ such that the following property holds:
\begin{equation}
T_R(C(x,y), C(x+1,y)) \wedge T_D(C(x,y), C(x,y+1)) \mbox{\ for each\ }x,y \in {\mathbb Z}. \label{tiling}
\end{equation}
However, no periodic tiling satisfying \ref{tiling} holds. A tiling $C: {\mathbb Z} \times {\mathbb Z} \rightarrow K$ is {\bf periodic} iff there exists
$(x_0, y_0) \neq (0,0)$ such that $C(x,y) = C(x+x_0, y+y_0)$ for each $x,y \in {\mathbb Z}$.
\end{theorem}
The original coloring by Berger used 20426 tiles. It is sufficient to use 11 tiles \cite{jeandel}.
We add a new relation $C$ for every tile $C \in K$. We also add the following axioms:
\begin{itemize}
\item {\bf Full tiling.} $\forall v \bigvee^!_{C \in K} C(v).$ Everything needs to have a color.
\item {\bf Correct tiling.}
For every pair of tiles $C_1, C_2 \in K$ such that $\neg T_R(C_1,C_2)$, we have $\neg \exists v C_1(v) \wedge C_2(R(v))$.
For every pair of tiles $C_1, C_2 \in K$ such that $\neg T_D(C_1,C_2)$, we have $\neg \exists v C_1(v) \wedge C_2(D(v))$.
\end{itemize}
Let $\phi_2$ be the conjuction of $\phi_1$ and the axioms above.
\begin{theorem}
If $G$ is a finite, connected model of $\phi_2$ then $G$ is a rectangular grid.
\end{theorem}
\begin{proof}
Take $v \in V$. If one of the relations $L$, $R$, $U$, $D$ is not defined for some $v \in V$, then $V$ is a rectangular grid by Theorem \ref{tgeo}.
Otherwise, let $C(x,y)$, for $x,y \geq 0$, be the relation $C \in K$ which is satisfied by $R^x(D^y(v))$. For $x<0$ or $y<0$, replace $R^x$ by $L^{-x}$ or $D^y$ by $U^{-y}$.
According to the correct tiling axiom, the property (\ref{tiling}) holds.
Since $V$ is finite, we must have $C(x_1,y_1)$ and $C(x_2,y_2)$ refer to the same element of our structure, even though $(x_1,y_1) \neq (x_2,y_2)$. It is easy to show that
$(x_1-x_2, y_1-y_2)$ is then the period of the tiling $C$, which contradicts Theorem \ref{wangth}.
$\rule{2mm}{2mm}$\par
\end{proof}
\begin{theorem}\label{allgrids}
There exists a formula $\phi_3$ such that the models of $\phi$, restricted to relations $L$, $R$, $U$, $D$, are precisely the rectangular grids $x^* \times y^*$ such that $y^* \leq 2^{x^*-1}$ and $x^* \leq 2^{y^*-1}$.
\end{theorem}
\begin{proof}
By adding an axiom that there exists exactly one element $v^*$ such that $L(v^*)$ and $U(v^*)$ are not defined, we obtain a formula $\phi_3$ whose all finite models are rectangular grids.
Now, take an $x^* \times y^*$ rectangular grid G. From Theorem \ref{wangth} there exists a tiling $C: {\mathbb Z} \times {\mathbb Z} \rightarrow K$ satisfying \ref{tiling}. Assign the relation $C(x,y)$
to each $(x,y)\in G$. If $y^* \leq 2^{x^*-1}$ and $x^* \leq 2^{y^*-1}$, we can also set $B_H(x,y)$ iff $x$-th bit of $y$ is 1, and $B_V(x,y)$ iff $y$-th bit of $x$ is 1.
Such a model will satisfy $\phi_3$. Note that if $x^* > 2^{y^*-1}$ or $y^* > 2^{x^*-1}$, the respective overflow axiom will not be satisfied.
$\rule{2mm}{2mm}$\par
\end{proof}
The number 2 in the theorem above can be changed to an integer $b \geq 2$ by using $b$-ary counters instead of the binary ones. However:
\begin{theorem} \label{srem} \rm
There is no formula $\phi$ over a signature consisting of $L$, $R$, $U$, $D$, and possibly extra unary relations whose all models restricted to relations $L$, $R$, $U$, $D$ are
precisely all rectangular grids. Furthermore, there is no such $\phi$ such that all models of $\phi$ are rectangular grids, and
there exists $y*$ such that for every $x*$ a rectangular model $x^* \times y^*$ of $\phi$ exists.
\end{theorem}
\begin{proof}
We will be using Hanf's locality lemma \cite{hanf}.
Let a $r$-neighborhood of the vertex $v \in V$, $N_r(v)$ be the set of all vertices whose distance from $v$ is at most $r$.
Let a {\bf $r$-type} of the vertex $v$, $\tau(v)$, be the isomorphism type of $N_r(v)$. When we restrict to models of degree bounded by $d$,
there are only finitely many such types. Let $T_r$ be the set of all types.
Let $f_{r,M}(G): T \rightarrow \{0..M\}$ be the function that assigns to each type $\tau \in T$ the minimum of $M$ and the number of vertices of type $\tau$ in $G$.
\begin{theorem}[Hanf's locality lemma\cite{hanf}]\label{hanf}
Let $\phi$ be a FO formula. Then there exist numbers $r$ and $M$ such that, for each graph $G=(V,E)$, $G \models \phi$ depends only on $f_{r,M}$.
\end{theorem}
Let $\phi$ be a FO formula such that all models of $\phi$ are rectangular grids. Take $r$ and $M$ from Theorem \ref{hanf}.
Let the rectangular grid $G$ be a model of $phi$, where
$V(G) = \{0,\ldots,x^*\} \times \{0,\ldots,y^*\}$. Let $\tau(x)$ be the type of column $x$, i.e., $\tau(x) = (\tau(x,0), \ldots, \tau(x,y*))$.
For sufficiently large $x^*$ there will be $x_1$ and $x_2$ such that $\tau(x_1+i) = \tau(x_2+i)$ for $i=-r, \ldots, r$ and
such that $f_{r,M}(G)(\tau(x,y)) \geq M$ for every $x \in \{x_1, \ldots, x_2\}$. Construct a new structure $G'$ by adding a cylinder of dimensions
$(x_2-x_1) \times y*$ to $G$, i.e.,
$V(G') = V(G) \cup \{(1,x,y): x \in \{x_1, \ldots, x_2-1\}, y \in \{0, \ldots, y*\}$, $U(1,x,y) = (1,x,y-1)$, $D(1,x,y) = (1,x,y+1)$,
$R(1,x,y) = (1,x+1,y)$, $L(1,x,y) = (1,x-1,y)$, $R(1,x_2-1,y) = (1,x_1,y)$, $L(1,x_1,y) = (1,x_2-1,y)$, whenever the point on the right hand side exists,
and undefined otherwise. For every unary relation $U$ we have $U(1,x,y)$ iff $U(x,y)$. It is easy to verify that $\tau(1,x,y) = \tau(x,y)$,
and every of these types already appeared at least $M$ times, and thus from Theorem \ref{hanf}, $G' \models \phi$.
$\rule{2mm}{2mm}$\par
\end{proof}
| 2,479 | 7,338 |
en
|
train
|
0.42.2
|
\section{Forced Planar Spectra}
\begin{corollary}
Let $S \subseteq \mathbb{N}$ be a set such that there exists a non-deterministic Turing machine (or 1DCA) recognizing the set of binary representations of
elements of $S$ in time $t(n)$ and memory $s(n)$, where $t(n) \cdot s(n) \leq n$ and $t(n),s(n) \geq \Omega(log(n))$. Then there exists a first-order formula $\phi$ such
that all models of $\phi$ are planar graphs, and the set of cardinalities of models of $\phi$ is $S$.
\end{corollary}
\begin{proof}
Let $A \subseteq {\mathbb N}$, and let $M$ be a non-deterministic Turing machine or a non-deterministic 1DCA recognizing $A$ in time $t(n)$ and space $s(n)$ such that $t(n) \cdot s(n) \leq n$.
A non-deterministic 1DCA is $M = (\Sigma, R, F)$ where $R \subseteq \Sigma^4$, and the final symbol $F \in \Sigma$. It is defined similar to a Turing machine,
but where computations are performed in parallel on all the tape cells: if $t(x,y)$ is the content of the tape at position $x$ and time $y$, then
the relation $R(t(x-1,y), t(x,y), t(x+1,y), t(x,y+1))$ must hold. The 1DCA accepts when it writes the symbol $F$.
Let $u(n) = n-t(n)s(n)$; without loss of
generality we can assume $u(n) < t(n)$.
It is well known that a first order formula on a grid can be used to simulate a Turing machine (or 1DCA):
the bottom row is the initial tape, and our formula ensures that each other row above it is a correct successor of the row below it.
Let $n \in A$. We will construct a formula $\phi$ which will have a model consisting of:
\begin{itemize}
\item A rectangular grid $G' = \{0,\ldots,s\} \times \{0,\ldots,t\}$, where $t = t(n)-1$ and $s = s(n)-1$. The structure of the grid is given by relations $L$, $R$, $U$ and $D$
just as in Section \ref{gridax}; we also have all the auxiliary relations required by Theorem \ref{allgrids}.
\item $u(n)$ elements which are not in the grid. The relation $P$ will hold for all the extra elements and only for them. The relation $Q$ will hold only
for the elements $(0,t-i) \in G'$ where $i \leq u(n)-1$. The relation $B$ gives a bijection between elements $x$ such that $P(x)$, and the elements $x$ such that $Q(x)$.
\item Encoding of the number $n$.
We encode the number $n$ in the leftmost cells in the initial tape using two relations $D_n$ and $E_n$ in the following way:
$D_a(x,t)$ is the $x$-th digit of $n$, and the relation $E_a$ signifies the end of the encoding: $E_a(x,y) \Rightarrow E_a(x+1,t) \wedge \neg D_a(x,t)$ (if $(x+1,t)$ exists).
\item Similarly we encode the numbers $s$, $t$ and $u$.
\item An encoded run of $M$ which accepts the encoded value $n$ as the input.
\item An encoded run of an one-dimensional cellular automaton $M_2$ which verifies that the relation $n = (s+1) \times (t+1) + u$ holds for the encoded numbers.
A one-dimensional cellular automaton can add and multiply $k$-digit numbers in time $O(k)$ \cite{atrubin}, hence our space $s$ will be sufficient.
\item Our grid already has the binary representations of $s$ and $t$ computed as the relations $B_H$ and $B_V$. In the case of $B_V$ the
computed $t$ is already where we need it (we only need to define the relation $E_{t}$ in the straightforward way). In the case of $B_H$ the
computed $s$ is in the rightmost column, so we add extra wiring relations $W$ to move it to the beginning of the initial tape. In the case of $u$,
we need to compute the binary representation of the number of rows $i$ such that $Q(0,i)$; this can be computed in the same way as we have computed
the number of all rows (using the relation $B_U$ similar to $B_V$).
\end{itemize}
\def\subfig#1#2{
\begin{minipage}[b]{.5\textwidth}\includegraphics[width=\textwidth]{#1}\begin{center}#2\end{center}
\end{minipage}
}
\begin{figure}
\caption{Computing the size of our model.\label{fig}
\label{fig}
\end{figure}
Figure \ref{fig} shows the elements of our construction. In all the pictures, the small circles are the extra elements (where $P$ holds), and the other elements are
the grid; the thin lines represent the relation $B$, the thick lines represent the relations $U$, $D$, $L$ and $R$. In \ref{fig}a the black circles
represent $B_H \equiv D_{t}$ and gray boxes represent $E_{t}$. In \ref{fig}b the gray circles represent $Q$, black circles represent $B_U$ and gray circles
represent $E_U$. In \ref{fig}c the black circles represent $B_V$, while in \ref{fig}d the extra thick lines represent $W$, black circles represent $E_{s}$,
and gray boxes represent $E_{s}$.
The formula $\phi$ will be the conjuction of the following axioms:
\begin{itemize}
\item (1) $\phi_3$, restricted to elements for which $P$ does not hold. This requires that we indeed have a rectangular grid.
\item (2) Axiomatiziations of the Turing machine $M$.
\item (3) $B$ is a bijection.
\item (4) The set of elements satisfying $Q$ has the correct shape: $Q(v) \Rightarrow \neg P(v) \wedge (\neg\exists y L(x,y)) \wedge (\exists y D(x,y) \Rightarrow Q(y))$,
\item (5) Axiomatiziation of $B_U$, similar to the axiomatization of $B_V$, but where we add 1 only in the rows $y$ where $Q(0,y)$ holds.
\item (6) Axiomatiziation of the wiring $W$ moving $s$. The axioms are as follows:
$W(v,w) \wedge D_t(v) \Rightarrow D_t(w)$; $W(v,w) \Rightarrow W(w,v)$; every v is connected to either (a) only $R(v)$ and $L(v)$ is undefined, (b) only $L(v)$ and $R(v)$;
(c) only $L(v)$ and $D(v)$; (d) only $D(v)$ and $U(v)$; (e) only $U(v)$ and $D(v)$ is not defined; (f) nothing. Furthermore, in case (c), $L(D(v))$ must either
be also case (c) or the bottom left corner; $D_t(v) \iff B_H(v)$ whenever $L(v)$ is undefined; and the case (a) holds whenever $L(v)$ undefined, $D_t(v)$,
and $D(v)$ defined.
\item (7) $\forall v D_{t}(v) \iff V_H(y)$.
\item (8) For every encoded number $a$, $E_a(v) \rightarrow (\neg D_a(v) \wedge \exists w R(v,w) \rightarrow E_a(w)$.
\item (9) Axiomatiziations of the automaton $M_2$.
\end{itemize}
Our model satisfies all these axioms.
On the other hand, suppose that $\phi$ has a model $G$ of size $n$. By (1) this model constists of a rectangular grid and
a number of $u$ extra elements. By (3) and (4) the relation $Q$ is satisfied only for $u$ bottommost elements in the leftmost column.
By (5) the encoded number $u$ equals the number of these elements. By (6) and (7) the encoded numbers $s$ and $t$ equal the dimensions of
the grid. By (8) and (9) we know that the encoded number $n$ indeed equals the size of $G$. By (2) we know that $M$ accepts $n$, therefore
$n \in A$.
\end{proof}
\end{document}
| 2,223 | 7,338 |
en
|
train
|
0.43.0
|
\begin{document}
\title[Fractional Choquard equations with magnetic fields]{Concentration phenomena for a fractional Choquard equation with magnetic field}
\author[V. Ambrosio]{Vincenzo Ambrosio}
\address{Vincenzo AmbrosioH^{s}_{\e}fill\break\indent
Department of Mathematics H^{s}_{\e}fill\break\indent
EPFL SB CAMA H^{s}_{\e}fill\break\indent
Station 8 CH-1015 Lausanne, Switzerland}
\varepsilonmail{[email protected]}
\subjclass{Primary 35A15, 35R11; Secondary 45G05}
\date{}
\keywords{Fractional Choquard equation, fractional magnetic Laplacian, penalization method.}
\begin{abstract}
We consider the following nonlinear fractional Choquard equation
$$
\varepsilon^{2s}(-\Delta)^{s}_{A/\varepsilon} u + V(x)u = \varepsilon^{\mu-N}\left(\frac{1}{|x|^{\mu}}*F(|u|^{2})\right)f(|u|^{2})u \mbox{ in } \mathbb{R}^{N},
$$
where $\varepsilon>0$ is a parameter, $s\in (0, 1)$, $0<\mu<2s$, $N\geq 3$, $(-\Delta)^{s}_{A}$ is the fractional magnetic Laplacian, $A:\mathbb{R}^{N}\rightarrow \mathbb{R}^{N}$ is a smooth magnetic potential, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a positive potential with a local minimum and $f$ is a continuous nonlinearity with subcritical growth.
By using variational methods we prove the existence and concentration of nontrivial solutions for $\varepsilon>0$ small enough.
\varepsilonnd{abstract}
\maketitle
| 469 | 38,947 |
en
|
train
|
0.43.1
|
\section{Introduction}
\noindent
In this paper we investigate the existence and concentration of nontrivial solutions for the following nonlinear fractional Choquard equation
\begin{equation}\label{P}
\varepsilon^{2s}(-\Delta)^{s}_{A/\varepsilon} u + V(x)u = \varepsilon^{\mu-N}\left(\frac{1}{|x|^{\mu}}*F(|u|^{2})\right)f(|u|^{2})u \,\mbox{ in } \,\mathbb{R}^{N},
\varepsilonnd{equation}
where $\varepsilon>0$ is a parameter, $s\in (0,1)$, $N\geq 3$, $0<\mu<2s$, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a continuous potential and
$A:\mathbb{R}^{N}\rightarrow \mathbb{R}^{N}$ is a $C^{0, \alpha}$ magnetic potential, with $\alpha\in (0,1]$.
The nonlocal operator $(-\Delta)^{s}_{A}$ is the fractional magnetic Laplacian which may be defined for any $u:\mathbb{R}^{N}\rightarrow \mathbb{C}$ smooth enough by setting
$$
(-\Delta)^{s}_{A}u(x)= c_{N,s} P.V. \int_{\mathbb{R}^{N}} \frac{u(x)-u(y)e^{\imath A(\frac{x+y}{2})\cdot (x-y)}}{|x-y|^{N+2s}} \,dy \quad (x\in \mathbb{R}^{N}),
$$
where $c_{N, s}$ is a normalizing constant. This operator has been introduced in \cite{DS, I} with motivations falling into the framework of the general theory of L\'evy processes.
As showed in \cite{SV}, when $s\rightarrow 1$, the operator $(-\Delta)^{s}_{A}$ reduces to the magnetic Laplacian (see \cite{LaL, LL}) defined as
$$
\left(\frac{1}{\imath}\nabla-A\right)^{2}\!u= -\Delta u -\frac{2}{\imath} A(x) \cdot \nabla u + |A(x)|^{2} u -\frac{1}{\imath} u \dive(A(x)),
$$
which has been widely investigated by many authors: see \cite{AF, AFF, AS, Cingolani, CS, EL, K}.
Recently, many papers dealt with different fractional problems involving the operator $(-\Delta)^{s}_{A}$.
d'Avenia and Squassina \cite{DS} studied the existence of ground states solutions for some fractional magnetic problems via minimization arguments.
Pinamonti et al. \cite{PSV1, PSV2} obtained a magnetic counterpart of the Bourgain-Brezis-Mironescu formula and the MazÕya-Shaposhnikova formula respectively; see also \cite{NPSV} for related results.
Zhang et al. \cite{ZSZ} proved a multiplicity result for a fractional magnetic Schr\"odinger equation with critical growth.
In \cite{MPSZ} Mingqi et al. studied existence and multiplicity of solutions for a subcritical fractional Schr\"{o}dinger-Kirchhoff equation involving an external magnetic potential.
Fiscella et al. \cite{FPV} considered a fractional magnetic problem in a bounded domain proving the existence of at least two nontrivial weak solutions under suitable assumptions on the nonlinear term.
In \cite{AD} the author and d'Avenia used variational methods and Ljusternick-Schnirelmann theory to prove existence and multiplicity of nontrivial solutions for a fractional Schr\"{o}dinger equation with subcritical nonlinearities. \\
We note that when $A=0$, the operator $(-\Delta)^{s}_{A}$ becomes the celebrated fractional Laplacian $(-\Delta)^{s}$ which arises in the study of several physical phenomena like phase transitions, crystal dislocations, quasi-geostrophic flows, flame propagations and so on. Due to the extensive literature on this topic, we refer the interested reader to \cite{DPMV, DPV, MBRS} and the references therein.\\
In absence of the magnetic field, equation \varepsilonqref{P} is a fractional Choquard equation of the type
\begin{equation}\label{FChE}
(-\Delta)^{s} u + V(x)u = \left(\frac{1}{|x|^{\mu}}*F(u)\right)f(u) \,\mbox{ in } \,\mathbb{R}^{N}.
\varepsilonnd{equation}
d'Avenia et al. \cite{DSS} studied the existence, regularity and asymptotic behavior of solutions to \varepsilonqref{FChE} when $f(u)=u^{p}$ and $V(x)\varepsilonquiv const$. If $V(x)=1$ and $f$ satisfies Berestycki-Lions type assumptions, the existence of ground state solutions for a fractional Choquard equation has been established in \cite{SGY}.
The analyticity and radial symmetry of positive ground state for a critical boson star equation has been considered by Frank and Lenzmann in \cite{FL}.
Recently, the author in \cite{Apota} studied the multiplicity and concentration of positive solutions for a fractional Choquard equation under local conditions on the potential $V(x)$. \\
When $s=1$, equation \varepsilonqref{FChE} reduces to the generalized Choquard equation:
\begin{equation}\label{GCE}
-\Delta u + V(x) u = \left(\frac{1}{|x|^{\mu}}*F(u)\right)f(u) \,\mbox{ in } \,\mathbb{R}^{N}.
\varepsilonnd{equation}
If $p=\mu=2$, $V(x)\varepsilonquiv 1$, $F(u)=\frac{u^{2}}{2}$ and $N=3$, \varepsilonqref{GCE} is called the Choquard-Pekar equation which goes back to the 1954's work by Pekar \cite{Pek} to the description of a polaron at rest in Quantum Field Theory and to 1976's model of Choquard of an electron trapped in its own hole as an approximation to Hartree-Fock theory for a one-component plasma \cite{LS}. The same equation was proposed by Penrose \cite{Pen} as a model of self-gravitating matter and is known in that context as the Schr\"odinger-Newton equation.
Lieb in \cite{Lieb} proved the existence and uniqueness of positive solutions to a Choquard-Pekar equation.
Subsequently, Lions \cite{Lions} established a multiplicity result via variational methods. Ackermann in \cite{Ack} proved the existence and multiplicity of solutions for \varepsilonqref{GCE} when $V$ is periodic.
Ma and Zhao \cite{MZ} showed that, up to translations, positive solutions of equation \varepsilonqref{GCE} with $f(u)=u^{p}$, are radially symmetric and monotone decreasing for suitable values of $\mu$, $N$ and $p$. This results has been improved by Moroz and Van Schaftingen in \cite{MVS1}.
The same authors in \cite{MVS2} obtained the existence of ground state solutions with a general nonlinearity $f$.
Cingolani et al. \cite{CSS} showed the existence of multi-bump type solutions for a Schro\"odinger equation in presence of electric and magnetic potentials and Hartree-type nonlinearities.
Alves et al. \cite{AFY}, inspired by \cite{AFF, CSS}, studied the multiplicity and concentration phenomena of solutions for \varepsilonqref{GCE} in presence of a magnetic field. For a more detailed bibliography on the Choquard equation we refer to \cite{MVS3}.
| 1,880 | 38,947 |
en
|
train
|
0.43.2
|
Motivated by \cite{AFY, Apota, AD}, in this paper we focus our attention on the existence and concentration of solutions to \varepsilonqref{P} under local conditions on the potential $V$. Before stating our main result, we introduce the assumptions on $V$ and $f$.
Along the paper, we assume that the potential $V: \mathbb{R}^{N}\rightarrow \mathbb{R}$ is a continuous function verifying the following conditions introduced in \cite{DPF}:
\begin{enumerate}
\item [$(V_1)$] $V(x)\geq V_{0}>0$ for all $x\in \mathbb{R}^{N}$;
\item [$(V_2)$] there exists a bounded open set $\Lambda\subset \mathbb{R}^{N}$ such that
$$
V_{0}=\inf_{x\in \Lambda} V(x)<\min_{x\in \partial \Lambda} V(x),
$$
\varepsilonnd{enumerate}
and $f: \mathbb{R}\rightarrow \mathbb{R}$ is a continuous function such that $f(t)=0$ for $t<0$ and satisfies the following assumptions:
\begin{enumerate}
\item [($f_1$)] $\displaystyle{\lim_{t\rightarrow 0} f(t)=0}$;
\item [($f_2$)] there exists $q\in (2, \frac{2^{*}_{s}}{2}(2-\frac{\mu}{N}))$, where $2^{*}_{s}=\frac{2N}{N-2s}$, such that $\displaystyle{\lim_{t\rightarrow \infty} \frac{f(t)}{t^{\frac{q-2}{2}}}=0}$;
\item [($f_3$)] the map $\displaystyle{t \mapsto f(t)}$ is increasing for every $t>0$.
\varepsilonnd{enumerate}
We point to that the restriction on $q$ in $(f_2)$ is related to the Hardy-Littlewood-Sobolev inequality:
\begin{theorem}\label{HLS}\cite{LL}
Let $r, t>1$ and $0<\mu<N$ such that $\frac{1}{r}+\frac{\mu}{N}+\frac{1}{t}=2$. Let $f\in L^{r}(\mathbb{R}^{N})$ and $h\in L^{t}(\mathbb{R}^{N})$. Then there exists a sharp constant $C(r, N, \mu, t)>0$ independent of $f$ and $h$ such that
$$
\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}} \frac{f(x)h(y)}{|x-y|^{\mu}}\, dx dy\leq C(r, N, \mu, t)\|f\|_{L^{r}(\mathbb{R}^{N})}\|h\|_{L^{t}(\mathbb{R}^{N})}.
$$
\varepsilonnd{theorem}
Indeed, by $(f_1)$ and $(f_2)$ it follows that $|F(|u|^{2})|\leq C(|u|^{2}+|u|^{q})$, so it is easy to check that
the term
\begin{equation}\label{chiara}
\left|\int_{\mathbb{R}^{N}} \left( \frac{1}{|x|^{\mu}}* F(|u|^{2})\right) F(|u|^{2}) dx\right|<\infty \quad \forall u\in H^{s}_{\e},
\varepsilonnd{equation}
where $H^{s}_{\e}$ is defined in Section $2$, when $F(|u|^{2})\in L^{t}(\mathbb{R}^{N})$ for all $t>1$ such that
$$
\frac{2}{t}+\frac{\mu}{N}=2, \, \mbox{ that is } t=\frac{2N}{2N-\mu}.
$$
Therefore, if $q\in (2, \frac{2^{*}_{s}}{2}(2-\frac{\mu}{N}))$ and $\mu\in (0, 2s)$ we can use the fractional Sobolev embedding $H^{s}(\mathbb{R}^{N}, \mathbb{R})\subset L^{r}(\mathbb{R}^{N}, \mathbb{R})$ for all $r\in [2, 2^{*}_{s}]$, to deduce that $tq\in (2, 2^{*}_{s})$ and then \varepsilonqref{chiara} holds true. \\
Now, we can state the main result of this paper:
\begin{theorem}\label{thm1}
Suppose that $V$ verifies $(V_1)$-$(V_2)$, $0<\mu<2s$ and $f$ satisfies $(f_1)$-$(f_3)$ with $q\in (2, 2\frac{(N-\mu)}{N-2s} )$. Then there exists $\varepsilon_{0}>0$ such that, for any $\varepsilon\in (0, \varepsilon_{0})$, problem \varepsilonqref{P} has a nontrivial solution. Moreover, if $|u_{\varepsilon}|$ denotes one of these solutions and $x_{\varepsilon}\in \mathbb{R}^{N}$ its global maximum, then
$$
\lim_{\varepsilon\rightarrow 0} V(x_{\varepsilon})=V_{0},
$$
and
\begin{align*}
|u_{\varepsilon}(x)|\leq \frac{\tilde{C} \varepsilon^{N+2s}}{\varepsilon^{N+2s}+|x-x_{\varepsilon}|^{N+2s}} \quad \forall x\in \mathbb{R}^{N}.
\varepsilonnd{align*}
\varepsilonnd{theorem}
\begin{remark}
Assuming $f\in C^{1}$, one can use Ljusternick-Schnirelmann theory and argue as in \cite{Apota, AD} to relate the number of nontrivial solutions to \varepsilonqref{P} with the topology of the set where the potential attains its minimum value.
\varepsilonnd{remark}
The proof of Theorem \ref{thm1} is inspired by some variational arguments used in \cite{AFF, AFY, AM, Apota}. Anyway, the presence of the fractional magnetic Laplacian and nonlocal Hartree-type nonlinearity does not permit to easily adapt in our setting the techniques developed in the above cited papers and, as explained in what follows, a more intriguing and accurate analysis will be needed. Firstly, after a change of variable, it is easy to check that problem (\ref{P}) is equivalent to the following one:
\begin{equation}\label{R}
(-\Delta)^{s}_{A_{\varepsilon}} u + V_{\varepsilon}(x)u = \left(\frac{1}{|x|^{\mu}}*F(|u|^{2})\right)f(|u|^{2})u \mbox{ in } \mathbb{R}^{N}
\varepsilonnd{equation}
where $A_{\varepsilon}(x):=A(\varepsilon x)$ and $V_{\varepsilon}(x):=V(\varepsilon x)$.
In the spirit of \cite{DPF} (see also \cite{AFF, AM}), we modify the nonlinearity in a suitable way and we consider an auxiliary problem. We note that the restriction imposed on $\mu$ allows us to use the penalization technique.
Without loss of generality, along the paper we will assume that $0\in \Lambda$ and $V_{0}=V(0)=\inf_{x\in\mathbb{R}^{N}} V(x)$.
Now, we fix $\varepsilonll>0$ large enough, which will be determined later on, and let $a>0$ be the unique number such that $f(a)=\frac{V_{0}}{\varepsilonll}$. Moreover, we introduce the functions
$$
\tilde{f}(t):=
\begin{cases}
f(t)& \text{ if $t \leq a$} \\
\frac{V_{0}}{\varepsilonll} & \text{ if $t >a$},
\varepsilonnd{cases}
$$
and
$$
g(x, t):=\chi_{\Lambda}(x)f(t)+(1-\chi_{\Lambda}(x))\tilde{f}(t),
$$
where $\chi_{\Lambda}$ is the characteristic function on $\Lambda$, and we write $G(x, t)=\int_{0}^{t} g(x, \tau)\, d\tau$.\\
From assumptions $(f_1)$-$(f_3)$, it is easy to verify that $g$ fulfills the following properties:
\begin{enumerate}
\item [($g_1$)] $\displaystyle{\lim_{t\rightarrow 0} g(x, t)=0}$ uniformly in $x\in \mathbb{R}^{N}$;
\item [($g_2$)] $\displaystyle{\lim_{t\rightarrow \infty} \frac{g(x, t)}{t^{\frac{q-2}{2}}}=0}$ uniformly in $x\in \mathbb{R}^{N}$;
\item [($g_3$)] $(i)$ $0\leq G(x, t)< g(x, t)t$ for any $x\in \Lambda$ and $t>0$, and \\
$(ii)$ $0<G(x, t)\leq g(x, t)t\leq \frac{V_{0}}{\varepsilonll}t$ for any $x\in \mathbb{R}^{N}\setminus \Lambda$ and $t>0$,
\item [($g_4$)] $t\mapsto g(x, t)$ and $t\mapsto \frac{G(x, t)}{t}$ are increasing for all $x\in \mathbb{R}^{N}$ and $t>0$.
\varepsilonnd{enumerate}
Thus, we consider the following auxiliary problem
\begin{equation*}
(-\Delta)^{s}_{A_{\varepsilon}} u + V_{\varepsilon}(x)u = \left(\frac{1}{|x|^{\mu}}*G(\varepsilon x, |u|^{2})\right)g(\varepsilon x, |u|^{2})u \mbox{ in } \mathbb{R}^{N},
\varepsilonnd{equation*}
and in view of the definition of $g$, we are led to seek solutions $u$ of the above problem such that
\begin{equation}\label{ue}
|u(x)|<a \mbox{ for all } x\in \mathbb{R}^{N}\setminus \Lambda_{\varepsilon}, \quad \mbox{ where } \Lambda_{\varepsilon}:=\{x\in \mathbb{R}^{N}: \varepsilon x\in \Lambda\}.
\varepsilonnd{equation}
By using this penalization technique and establishing some careful estimates on the convolution term, we are able to prove that the energy functional associated with the auxiliary problem has a mountain pass geometry and satisfies the Palais-Smale condition; see Lemma \ref{MPG}, \ref{lemK} and \ref{PSc}. Then we can apply the Mountain Pass Theorem \cite{AR} to obtain the existence of a nontrivial solution $u_{\varepsilon}$ to the modified problem.
The H\"older regularity assumption on the magnetic field $A$ and the fractional diamagnetic inequality \cite{DS}, will be properly exploited to show an interesting and useful relation between the mountain pass minimax level $c_{\varepsilon}$ of the modified functional and the minimax level $c_{V_{0}}$ associated with the limit functional; see Lemma \ref{AMlem1}.
In order to verify that $u_{\varepsilon}$ is also solution of the original problem \varepsilonqref{P}, we need to check that $u_{\varepsilon}$ verifies \varepsilonqref{ue} for $\varepsilon>0$ sufficiently small.
To achieve our goal, we first use an appropriate Moser iterative scheme \cite{Moser} to show that $\|u_{\varepsilon}\|_{L^{\infty}(\mathbb{R}^{N})}$ is bounded uniformly with respect to $\varepsilon$. In these estimates, we take care of the fact that the convolution term
is a bounded term in view of Lemma \ref{lemK}.
After that, we use these informations to develop a very clever approximation argument related in some sense to the following fractional version of Kato's inequality \cite{Kato}
$$
(-\Delta)^{s}|u|\leq \mathbb{R}e(sign(u)(-\Delta)^{s}_{A}u),
$$
to show that $|u_{\varepsilon}|$ is a weak subsolution to the problem
$$
(-\Delta)^{s}|u|+V(x)|u|= h(|u|^{2})|u| \mbox{ in } \mathbb{R}^{N},
$$
for some subcritical nonlinearity $h$, and then we prove that $|u_{\varepsilon}(x)|\rightarrow 0$ as $|x|\rightarrow \infty$, uniformly in $\varepsilon$; see Lemma \ref{moser}. We point out that our arguments are different from the ones used in the classical case $s=1$ and the fractional setting $s\in (0,1)$ without magnetic field. Indeed, we don't know if a Kato's inequality is available in our framework, so we can not proceed as in \cite{CS, K} in which the Kato's inequality is combined with some standard elliptic estimates to obtain informations on the decay of solutions. Moreover, the appearance of magnetic field $A$ and the nonlocal character of $(-\Delta)^{s}_{A}$ do not permit to adapt the iteration argument developed in \cite{AFF, AFY} where $s=1$ and $A\not \varepsilonquiv 0$, and we can not use the well-known estimates based on the Bessel kernel (see \cite{AM, FQT}) established for fractional Schr\"odinger equations with $A=0$.
However, we believe that the ideas contained here can be also applied to deal with other fractional magnetic problems like \varepsilonqref{P}. Finally, we also give an estimate on the decay of modulus of solutions to \varepsilonqref{P} which is in clear accordance with the results in \cite{FQT}.
To the best of our knowledge, this is the first time that the penalization method is used to study nontrivial solutions for fractional Choquard equations with magnetic fields, and this represents the novelty of this work.\\
The paper is organized as follows: in Section $2$ we present some preliminary results and we collect some useful lemmas.
The Section $3$ is devoted to the proof of Theorem \ref{thm1}.
| 3,328 | 38,947 |
en
|
train
|
0.43.3
|
\section{Preliminaries and functional setting}
\noindent
For any $s\in (0,1)$, we denote by $\mathcal{D}^{s, 2}(\mathbb{R}^{N}, \mathbb{R})$ the completion of $C^{\infty}_{0}(\mathbb{R}^{N}, \mathbb{R})$ with respect to
$$
[u]^{2}=\iint_{\mathbb{R}^{2N}} \frac{|u(x)-u(y)|^{2}}{|x-y|^{N+2s}} \, dx \, dy =\|(-\Delta)^{\frac{s}{2}} u\|^{2}_{L^{2}(\mathbb{R}^{N})},
$$
that is
$$
\mathcal{D}^{s, 2}(\mathbb{R}^{N}, \mathbb{R})=\left\{u\in L^{2^{*}_{s}}(\mathbb{R}^{N},\mathbb{R}): [u]_{H^{s}(\mathbb{R}^{N})}<\infty\right\}.
$$
Let us introduce the fractional Sobolev space
$$
H^{s}(\mathbb{R}^{N}, \mathbb{R})= \left\{u\in L^{2}(\mathbb{R}^{N}) : \frac{|u(x)-u(y)|}{|x-y|^{\frac{N+2s}{2}}} \in L^{2}(\mathbb{R}^{2N}) \right \}
$$
endowed with the natural norm
$$
\|u\| = \sqrt{[u]^{2} + \|u\|_{L^{2}(\mathbb{R}^{N})}^{2}}.
$$
Let us denote by $L^{2}(\mathbb{R}^{N}, \mathbb{C})$ the space of complex-valued functions with summable square, endowed with the real
scalar product
$$
\langle u, v\rangle_{L^{2}}=\mathbb{R}e\left(\int_{\mathbb{R}^{N}} u \bar{v} dx\right)
$$
for all $u, v\in L^{2}(\mathbb{R}^{N}, \mathbb{C})$.
We consider the space
$$
\mathcal{D}^{s}_{A}(\mathbb{R}^{N}, \mathbb{C})=\{u\in L^{2^{*}_{s}}(\mathbb{R}^{N}, \mathbb{C}) : [u]_{A}<\infty\}
$$
where
$$
[u]_{A}^{2}=\iint_{\mathbb{R}^{2N}} \frac{|u(x)-u(y)e^{\imath A(\frac{x+y}{2})\cdot (x-y)}|^{2}}{|x-y|^{N+2s}} dx dy.
$$
Then, we define the following fractional magnetic Sobolev space
$$
H^{s}_{A}(\mathbb{R}^{N}, \mathbb{C})=\{u\in L^{2}(\mathbb{R}^{N}, \mathbb{C}): [u]_{A}<\infty\}.
$$
It is easy to check that $H^{s}_{A}(\mathbb{R}^{N}, \mathbb{C})$ is a Hilbert space with the real scalar product
\begin{align*}
\langle u, v\rangle_{s, A}&=\mathbb{R}e\iint_{\mathbb{R}^{2N}} \frac{(u(x)-u(y)e^{\imath A(\frac{x+y}{2})\cdot (x-y)})\overline{(v(x)-v(y)e^{\imath A(\frac{x+y}{2})\cdot (x-y)})}}{|x-y|^{N+2s}} dx dy\\
&\quad +\langle u, v\rangle_{L^{2}}
\varepsilonnd{align*}
for any $u, v\in H^{s}_{A}(\mathbb{R}^{N}, \mathbb{C})$. Moreover, $C^{\infty}_{c}(\mathbb{R}^{N}, \mathbb{C})$ is dense in $H^{s}_{A}(\mathbb{R}^{N}, \mathbb{C})$ (see \cite{AD}).
Now, we recall the following useful results:
\begin{theorem}\label{Sembedding}\cite{DS}
The space $H^{s}_{A}(\mathbb{R}^{N}, \mathbb{C})$ is continuously embedded into $L^{r}(\mathbb{R}^{N}, \mathbb{C})$ for any $r\in [2, 2^{*}_{s}]$ and compactly embedded into $L^{r}(K, \mathbb{C})$ for any $r\in [1, 2^{*}_{s})$ and any compact $K\subset \mathbb{R}^{N}$.
\varepsilonnd{theorem}
\begin{lemma}\label{DI}\cite{DS}
For any $u\in H^{s}_{A}(\mathbb{R}^{N}, \mathbb{C})$, we get $|u|\in H^{s}(\mathbb{R}^{N},\mathbb{R})$ and it holds
$$
[|u|]\leq [u]_{A}.
$$
We also have the following pointwise diamagnetic inequality
$$
||u(x)|-|u(y)||\leq |u(x)-u(y)e^{\imath A(\frac{x+y}{2})\cdot (x-y)}| \mbox{ a.e. } x, y\in \mathbb{R}^{N}.
$$
\varepsilonnd{lemma}
\begin{lemma}\label{aux}\cite{AD}
If $u\in H^{s}(\mathbb{R}^{N}, \mathbb{R})$ and $u$ has compact support, then $w=e^{\imath A(0)\cdot x} u \in H^{s}_{A}(\mathbb{R}^{N}, \mathbb{C})$.
\varepsilonnd{lemma}
\noindent
For any $\varepsilon>0$, we denote by
$$
H^{s}_{\e}=\left\{u\in \mathcal{D}^{s}_{A_{\varepsilon}}(\mathbb{R}^{N}, \mathbb{C}): \int_{\mathbb{R}^{N}} V_{\varepsilon}(x) |u|^{2}\, dx<\infty\right\}
$$
endowed with the norm
$$
\|u\|^{2}_{\varepsilon}=[u]^{2}_{A_{\varepsilon}}+\|\sqrt{V_{\varepsilon}} |u|\|^{2}_{L^{2}(\mathbb{R}^{N})}.
$$
\noindent
From now on, we consider the following auxiliary problem
\begin{equation}\label{Pe}
(-\Delta)^{s}_{A_{\varepsilon}} u + V_{\varepsilon}(x)u = \left(\frac{1}{|x|^{\mu}}*G(\varepsilon x, |u|^{2})\right)g(\varepsilon x, |u|^{2})u \mbox{ in } \mathbb{R}^{N}
\varepsilonnd{equation}
and we note that if $u$ is a solution of (\ref{Pe}) such that
\begin{equation*}
|u(x)|<a \mbox{ for all } x\in \mathbb{R}^{N}\setminus \Lambda_{\varepsilon},
\varepsilonnd{equation*}
then $u$ is indeed solution of the original problem (\ref{R}).
It is clear that weak solutions to (\ref{Pe}) can be found as critical points of the Euler-Lagrange functional $J_{\varepsilon}: H^{s}_{\varepsilon}\rightarrow \mathbb{R}$ defined by
$$
J_{\varepsilon}(u)=\frac{1}{2}\|u\|^{2}_{\varepsilon}-\frac{1}{4}\int_{\mathbb{R}^{N}} \left(\frac{1}{|x|^{\mu}}*G(\varepsilon x, |u|^{2})\right)G(\varepsilon x, |u|^{2})\, dx.
$$
We begin proving that $J_{\varepsilon}$ possesses a mountain pass geometry \cite{AR}.
\begin{lemma}\label{MPG}
$J_{\varepsilon}$ has a mountain pass geometry, that is
\begin{enumerate}
\item [$(i)$] there exist $\alpha, \rho>0$ such that $J_{\varepsilon}(u)\geq \alpha$ for any $u\in H^{s}_{\varepsilon}$ such that $\|u\|_{\varepsilon}=\rho$;
\item [$(ii)$] there exists $e\in H^{s}_{\varepsilon}$ with $\|e\|_{\varepsilon}>\rho$ such that $J_{\varepsilon}(e)<0$.
\varepsilonnd{enumerate}
\varepsilonnd{lemma}
\begin{proof}
By using $(g_1)$ and $(g_2)$ we know that for any $\varepsilonta>0$ there exists $C_{\varepsilonta}>0$ such that
\begin{equation}\label{g-estimate}
|g(\varepsilon x, t)|\leq \varepsilonta +C_{\varepsilonta} |t|^{\frac{q-2}{2}}.
\varepsilonnd{equation}
In view of Theorem \ref{HLS} and \varepsilonqref{g-estimate}, we can deduce that
\begin{align}\label{a1}
\left|\int_{\mathbb{R}^{N}} \left(\frac{1}{|x|^{\mu}}*G(\varepsilon x, |u|^{2})\right)G(\varepsilon x, |u|^{2})\, dx\right| \leq C\left(\int_{\mathbb{R}^{N}} (|u|^{2}+|u|^{q}\, dx)^{t}\right)^{\frac{2}{t}},
\varepsilonnd{align}
where $\frac{1}{t}=\frac{1}{2}(2-\frac{\mu}{N})$. Since $2<q<\frac{2^{*}_{s}}{2}(2-\frac{\mu}{N})$ we have $t q\in (2, 2^{*}_{s})$ and by using Theorem \ref{Sembedding} we can see that
\begin{align}\label{a2}
\left(\int_{\mathbb{R}^{N}} (|u|^{2}+|u|^{q}\, dx)^{t}\right)^{\frac{2}{t}}\leq C(\|u\|^{2}_{\varepsilon}+\|u\|^{q}_{\varepsilon})^{2}.
\varepsilonnd{align}
Putting together \varepsilonqref{a1} and \varepsilonqref{a2} we get
\begin{align*}
\left|\int_{\mathbb{R}^{N}} \left(\frac{1}{|x|^{\mu}}*G(\varepsilon x, |u|^{2})\right)G(\varepsilon x, |u|^{2})\, dx\right|\leq C(\|u\|^{2}_{\varepsilon}+\|u\|^{q}_{\varepsilon})^{2}\leq C(\|u\|^{4}_{\varepsilon}+\|u\|^{2q}_{\varepsilon}).
\varepsilonnd{align*}
Hence
$$
J(u)\geq \frac{1}{2}\|u\|^{2}_{\varepsilon}-C(\|u\|^{4}_{\varepsilon}+\|u\|^{2q}_{\varepsilon}),
$$
and recalling that $q>2$ we can infer that $(i)$ is satisfied.\\
Now, take a nonnegative function $u_{0}\in H^{s}(\mathbb{R}^{N}, \mathbb{R})\setminus\{0\}$ with compact support such that $supp(u_{0})\subset \Lambda_{\varepsilon}$. Then, by Lemma \ref{aux} we know that $u_{0}(x)e^{\imath A(0)\cdot x}\in H^{s}_{\varepsilon}\setminus\{0\}$. Set
$$
h(t)=\mathfrak{F}\left(\frac{t u_{0}}{\|u_{0}\|_{\varepsilon}}\right) \mbox{ for } t>0,
$$
where
$$
\mathfrak{F}(u)=\frac{1}{4}\int_{\mathbb{R}^{N}} \left( \frac{1}{|x|^{\mu}}*F(|u|^{2}) \right) F(|u|^{2})\,dx.
$$
From $(f_3)$ we know that $F(t)\leq f(t)t$ for all $t>0$.
Then, being $G(\varepsilon x, |u_{0}|^{2})=F(|u_{0}|^{2})$, we deduce that
\begin{align}\label{a3}
\frac{h'(t)}{h(t)}\geq \frac{4}{t} \quad \forall t>0.
\varepsilonnd{align}
Integrating \varepsilonqref{a3} over $[1, t\|u_{0}\|_{\varepsilon}]$ with $t>\frac{1}{\|u_{0}\|_{\varepsilon}}$, we get
$$
\mathfrak{F}(t u_{0})\geq \mathfrak{F}\left(\frac{u_{0}}{\|u_{0}\|_{\varepsilon}}\right) \|u_{0}\|_{\varepsilon}^{4}t^{4}.
$$
Summing up
$$
J_{\varepsilon}(t u_{0})\leq C_{1} t^{2}-C_{2}t^{4} \mbox{ for } t>\frac{1}{\|u_{0}\|_{\varepsilon}}.
$$
Taking $e=t u_{0}$ with $t$ sufficiently large, we can see that $(ii)$ holds.
\varepsilonnd{proof}
| 3,042 | 38,947 |
en
|
train
|
0.43.4
|
It is clear that weak solutions to (\ref{Pe}) can be found as critical points of the Euler-Lagrange functional $J_{\varepsilon}: H^{s}_{\varepsilon}\rightarrow \mathbb{R}$ defined by
$$
J_{\varepsilon}(u)=\frac{1}{2}\|u\|^{2}_{\varepsilon}-\frac{1}{4}\int_{\mathbb{R}^{N}} \left(\frac{1}{|x|^{\mu}}*G(\varepsilon x, |u|^{2})\right)G(\varepsilon x, |u|^{2})\, dx.
$$
We begin proving that $J_{\varepsilon}$ possesses a mountain pass geometry \cite{AR}.
\begin{lemma}\label{MPG}
$J_{\varepsilon}$ has a mountain pass geometry, that is
\begin{enumerate}
\item [$(i)$] there exist $\alpha, \rho>0$ such that $J_{\varepsilon}(u)\geq \alpha$ for any $u\in H^{s}_{\varepsilon}$ such that $\|u\|_{\varepsilon}=\rho$;
\item [$(ii)$] there exists $e\in H^{s}_{\varepsilon}$ with $\|e\|_{\varepsilon}>\rho$ such that $J_{\varepsilon}(e)<0$.
\varepsilonnd{enumerate}
\varepsilonnd{lemma}
\begin{proof}
By using $(g_1)$ and $(g_2)$ we know that for any $\varepsilonta>0$ there exists $C_{\varepsilonta}>0$ such that
\begin{equation}\label{g-estimate}
|g(\varepsilon x, t)|\leq \varepsilonta +C_{\varepsilonta} |t|^{\frac{q-2}{2}}.
\varepsilonnd{equation}
In view of Theorem \ref{HLS} and \varepsilonqref{g-estimate}, we can deduce that
\begin{align}\label{a1}
\left|\int_{\mathbb{R}^{N}} \left(\frac{1}{|x|^{\mu}}*G(\varepsilon x, |u|^{2})\right)G(\varepsilon x, |u|^{2})\, dx\right| \leq C\left(\int_{\mathbb{R}^{N}} (|u|^{2}+|u|^{q}\, dx)^{t}\right)^{\frac{2}{t}},
\varepsilonnd{align}
where $\frac{1}{t}=\frac{1}{2}(2-\frac{\mu}{N})$. Since $2<q<\frac{2^{*}_{s}}{2}(2-\frac{\mu}{N})$ we have $t q\in (2, 2^{*}_{s})$ and by using Theorem \ref{Sembedding} we can see that
\begin{align}\label{a2}
\left(\int_{\mathbb{R}^{N}} (|u|^{2}+|u|^{q}\, dx)^{t}\right)^{\frac{2}{t}}\leq C(\|u\|^{2}_{\varepsilon}+\|u\|^{q}_{\varepsilon})^{2}.
\varepsilonnd{align}
Putting together \varepsilonqref{a1} and \varepsilonqref{a2} we get
\begin{align*}
\left|\int_{\mathbb{R}^{N}} \left(\frac{1}{|x|^{\mu}}*G(\varepsilon x, |u|^{2})\right)G(\varepsilon x, |u|^{2})\, dx\right|\leq C(\|u\|^{2}_{\varepsilon}+\|u\|^{q}_{\varepsilon})^{2}\leq C(\|u\|^{4}_{\varepsilon}+\|u\|^{2q}_{\varepsilon}).
\varepsilonnd{align*}
Hence
$$
J(u)\geq \frac{1}{2}\|u\|^{2}_{\varepsilon}-C(\|u\|^{4}_{\varepsilon}+\|u\|^{2q}_{\varepsilon}),
$$
and recalling that $q>2$ we can infer that $(i)$ is satisfied.\\
Now, take a nonnegative function $u_{0}\in H^{s}(\mathbb{R}^{N}, \mathbb{R})\setminus\{0\}$ with compact support such that $supp(u_{0})\subset \Lambda_{\varepsilon}$. Then, by Lemma \ref{aux} we know that $u_{0}(x)e^{\imath A(0)\cdot x}\in H^{s}_{\varepsilon}\setminus\{0\}$. Set
$$
h(t)=\mathfrak{F}\left(\frac{t u_{0}}{\|u_{0}\|_{\varepsilon}}\right) \mbox{ for } t>0,
$$
where
$$
\mathfrak{F}(u)=\frac{1}{4}\int_{\mathbb{R}^{N}} \left( \frac{1}{|x|^{\mu}}*F(|u|^{2}) \right) F(|u|^{2})\,dx.
$$
From $(f_3)$ we know that $F(t)\leq f(t)t$ for all $t>0$.
Then, being $G(\varepsilon x, |u_{0}|^{2})=F(|u_{0}|^{2})$, we deduce that
\begin{align}\label{a3}
\frac{h'(t)}{h(t)}\geq \frac{4}{t} \quad \forall t>0.
\varepsilonnd{align}
Integrating \varepsilonqref{a3} over $[1, t\|u_{0}\|_{\varepsilon}]$ with $t>\frac{1}{\|u_{0}\|_{\varepsilon}}$, we get
$$
\mathfrak{F}(t u_{0})\geq \mathfrak{F}\left(\frac{u_{0}}{\|u_{0}\|_{\varepsilon}}\right) \|u_{0}\|_{\varepsilon}^{4}t^{4}.
$$
Summing up
$$
J_{\varepsilon}(t u_{0})\leq C_{1} t^{2}-C_{2}t^{4} \mbox{ for } t>\frac{1}{\|u_{0}\|_{\varepsilon}}.
$$
Taking $e=t u_{0}$ with $t$ sufficiently large, we can see that $(ii)$ holds.
\varepsilonnd{proof}
\noindent
Denoting by $c_{\varepsilon}$ the mountain pass level of the functional $J_{\varepsilon}$ and recalling that $supp(u_{0})\subset \Lambda_{\varepsilon}$, we can find $\kappa>0$ independent of $\varepsilon, l, a$ such that
$$
c_{\varepsilon}= \inf_{u\in H^{s}_{\varepsilon}\setminus \{0\}}\max_{t\geq 0} J_{\varepsilon}(tu) <\kappa
$$
for all $\varepsilon>0$ small.
Now, let us define
$$
\B=\{u\in H^{s}(\mathbb{R}^{N}): \|u\|^{2}_{\varepsilon}\leq 4(\kappa+1)\}
$$
and we set
$$
\tilde{K}_{\varepsilon}(u)(x)=\frac{1}{|x|^{\mu}}*G(\varepsilon x, |u|^{2}).
$$
The next lemma is very useful because allows us to treat the convolution term as a bounded term.
\begin{lemma}\label{lemK}
Assume that $(f_1)$-$(f_3)$ hold and $2<q<\frac{2(N-\mu)}{N-2s}$. Then there exists $\varepsilonll_{0}>0$ such that
$$
\frac{\sup_{u\in \B} \|\tilde{K}_{\varepsilon}(u)(x)\|_{L^{\infty}(\mathbb{R}^{N})}}{\varepsilonll_{0}}<\frac{1}{2} \mbox{ for any } \varepsilon>0.
$$
\varepsilonnd{lemma}
\begin{proof}
Let us prove that there exists $C_{0}>0$ such that
\begin{equation}\label{a6}
\sup_{u\in \B} \|\tilde{K}_{\varepsilon}(u)(x)\|_{L^{\infty}(\mathbb{R}^{N})}\leq C_{0}.
\varepsilonnd{equation}
First of all, we can observe that
\begin{equation}\label{a5}
|G(\varepsilon x, |u|^{2})|\leq |F(|u|^{2})|\leq C(|u|^{2}+|u|^{q}) \mbox{ for all } \varepsilon>0.
\varepsilonnd{equation}
Hence, by using \varepsilonqref{a5}, we can see that
\begin{align}\label{a7}
|\tilde{K}_{\varepsilon}(u)(x)|
&\leq \Bigl| \int_{|x-y|\leq 1} \frac{F(|u|^{2})}{|x-y|^{\mu}} \,dy\Bigr|+\Bigl| \int_{|x-y|>1} \frac{F(|u|^{2})}{|x-y|^{\mu}} \,dy\Bigr| \nonumber\\
&\leq C \int_{|x-y|\leq 1} \frac{|u(y)|^{2}+|u(y)|^{q}}{|x-y|^{\mu}}\, dy+C \int_{\mathbb{R}^{N}} (|u|^{2}+|u|^{q})\, dy \nonumber\\
&\leq C \int_{|x-y|\leq 1} \frac{|u(y)|^{2}+|u(y)|^{q}}{|x-y|^{\mu}}\, dy+C
\varepsilonnd{align}
where in the last line we used Theorem \ref{Sembedding} and $\|u\|^{2}_{\varepsilon}\leq 4(\kappa+1)$.\\
Now, we take
$$
t\in \Bigl(\frac{N}{N-\mu}, \frac{N}{N-2s}\Bigr] \mbox{ and } r\in \Bigl(\frac{N}{N-\mu}, \frac{2N}{q(N-2s)}\Bigr].
$$
By applying H\"older inequality, Theorem \ref{Sembedding} and $\|u\|^{2}_{\varepsilon}\leq 4(\kappa+1)$ we get
\begin{align}\label{a8}
\int_{|x-y|\leq 1} \frac{|u(y)|^{2}}{|x-y|^{\mu}}\, dy&\leq \Bigl(\int_{|x-y|\leq 1} |u|^{2t}\, dy \Bigr)^{\frac{1}{t}} \Bigl(\int_{|x-y|\leq 1} \frac{1}{|x-y|^{\frac{t\mu}{t-1}}}\, dy \Bigr)^{\frac{t-1}{t}}\nonumber \\
&\leq C_{*}(4(\kappa+1))^{2} \Bigl(\int_{\rho\leq 1} \rho^{N-1-\frac{t \mu}{t-1}}\, d\rho \Bigr)^{\frac{t-1}{t}}<\infty,
\varepsilonnd{align}
because of $N-1-\frac{t \mu}{t-1}>-1$.
In similar fashion we can prove
\begin{align}\label{a9}
\int_{|x-y|\leq 1} \frac{|u(y)|^{q}}{|x-y|^{\mu}}\, dy&\leq \Bigl(\int_{|x-y|\leq 1} |u|^{rq}\, dy \Bigr)^{\frac{1}{r}} \Bigl(\int_{|x-y|\leq 1} \frac{1}{|x-y|^{\frac{r\mu}{r-1}}}\, dy \Bigr)^{\frac{r-1}{r}}\nonumber \\
&\leq C_{*}(4(\kappa+1))^{q} \Bigl(\int_{\rho\leq 1} \rho^{N-1-\frac{r \mu}{r-1}}\, d\rho \Bigr)^{\frac{r-1}{r}}<\infty
\varepsilonnd{align}
in view of $N-1-\frac{r \mu}{r-1}>-1$.
Putting together \varepsilonqref{a8} and \varepsilonqref{a9} we obtain
$$
\int_{|x-y|\leq 1} \frac{|u(y)|^{2}+|u(y)|^{q}}{|x-y|^{\mu}}\, dy\leq C \mbox{ for all } x\in \mathbb{R}^{N}
$$
which together with \varepsilonqref{a7} implies \varepsilonqref{a6}.
Then we can find $\varepsilonll_{0}>0$ such that
$$
\frac{\sup_{u\in \B} \|\tilde{K}_{\varepsilon}(u)(x)\|_{L^{\infty}(\mathbb{R}^{N})}}{\varepsilonll_{0}}\leq \frac{C_{0}}{\varepsilonll_{0}}< \frac{1}{2}.
$$
\varepsilonnd{proof}
\noindent
Let $\varepsilonll_{0}$ be as in Lemma \ref{lemK} and $a>0$ be the unique number such that
$$
f(a)=\frac{V_{0}}{\varepsilonll_{0}}.
$$
From now on we consider the penalized problem \varepsilonqref{Pe} with these choices.
| 3,039 | 38,947 |
en
|
train
|
0.43.5
|
\noindent
Let $\varepsilonll_{0}$ be as in Lemma \ref{lemK} and $a>0$ be the unique number such that
$$
f(a)=\frac{V_{0}}{\varepsilonll_{0}}.
$$
From now on we consider the penalized problem \varepsilonqref{Pe} with these choices.
\noindent
In what follows, we show that $J_{\varepsilon}$ verifies a local compactness condition.
\begin{lemma}\label{PSc}
$J_{\varepsilon}$ satisfies the $(PS)_{c}$ condition for all $c\in [c_{\varepsilon}, \kappa]$.
\varepsilonnd{lemma}
\begin{proof}
Let $(u_{n})$ be a Palais-Smale sequence at the level $c$, that is $J_{\varepsilon}(u_{n})\rightarrow c$ and $J_{\varepsilon}'(u_{n})\rightarrow 0$. Let us note that $(u_{n})$ is bounded and there exists $n_{0}\in \mathbb{N}$ such that $\|u_{n}\|^{2}_{\varepsilon}\leq 4(\kappa+1)$ for all $n\geq n_{0}$. Indeed, by using $(g_3)$ and Lemma \ref{lemK}, we can see that
\begin{align*}
c+o_{n}(1)\|u_{n}\|_{\varepsilon}\geq J_{\varepsilon}(u_{n})-\frac{1}{4}\langle J'_{\varepsilon}(u_{n}), u_{n}\rangle\geq \frac{1}{4} \|u_{n}\|^{2}_{\varepsilon}
\varepsilonnd{align*}
which implies the thesis.\\
Now, we divide the proof in two main steps.\\
{\bf Step $1$}: For any $\varepsilonta>0$ there exists $R=R_{\varepsilonta}>0$ such that
\begin{equation}\label{DF}
\limsup_{n\rightarrow \infty}\int_{\mathbb{R}^{N}\setminus B_{R}} \int_{\mathbb{R}^{N}} \frac{|u_{n}(x)-u_{n}(y)e^{\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)}|^{2}}{|x-y|^{N+2s}}dx dy+\int_{\mathbb{R}^{N}\setminus B_{R}}V(\varepsilon x)|u_{n}|^{2}\, dx<\varepsilonta.
\varepsilonnd{equation}
Since $(u_{n})$ is bounded in $H^{s}_{\e}$, we may assume that $u_{n}\rightharpoonup u$ in $H^{s}_{\e}$ and $|u_{n}|\rightarrow |u|$ in $L^{r}_{loc}(\mathbb{R}^{N})$ for any $r\in [2, 2^{*}_{s})$.
Moreover, by Lemma \ref{lemK}, we can deduce that
\begin{equation}\label{Kbound}
\frac{\sup_{n\geq n_{0}}\|\tilde{K}_{\varepsilon}(u_{n})(x)\|_{L^{\infty}(\mathbb{R}^{N})}}{\varepsilonll_{0}}\leq \frac{1}{2}.
\varepsilonnd{equation}
Fix $R>0$ and let $\psi_{R}\in C^{\infty}(\mathbb{R}^{N})$ be a function such that $\psi_{R}=0$ in $B_{R/2}$, $\psi_{R}=1$ in $B_{R}^{c}$, $\psi_{R}\in [0, 1]$ and $|\nabla \varepsilonta_{R}|\leq C/R$.
Since $\langle J'_{\varepsilon}(u_{n}), \varepsilonta_{R}u_{n}\rangle =o_{n}(1)$ we have
\begin{align*}
&\mathbb{R}e\Bigl(\iint_{\mathbb{R}^{2N}} \!\!\!\frac{(u_{n}(x)\!-\!u_{n}(y)e^{\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)})\overline{((u_{n}\varepsilonta_{R})(x)\!-\!(u_{n}\varepsilonta_{R})(y)e^{\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)})}}{|x-y|^{N+2s}} dx dy \Bigr)\\
&+\int_{\mathbb{R}^{N}} V_{\varepsilon}(x)\varepsilonta_{R} |u_{n}|^{2}\, dx=\int_{\mathbb{R}^{N}} \Bigl(\frac{1}{|x|^{\mu}}*G(\varepsilon x, |u_{n}|^{2})\Bigr) g(\varepsilon x, |u_{n}|^{2})u_{n}\psi_{R}+o_{n}(1).
\varepsilonnd{align*}
Taking into account
\begin{align*}
&\mathbb{R}e\Bigl(\iint_{\mathbb{R}^{2N}}\!\!\! \frac{(u_{n}(x)\!-\!u_{n}(y)e^{\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)})\overline{((u_{n}\varepsilonta_{R})(x)\!-\!(u_{n}\varepsilonta_{R})(y)e^{\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)})}}{|x-y|^{N+2s}} dx dy \Bigr)\\
&=\mathbb{R}e\Bigl(\iint_{\mathbb{R}^{2N}} \!\!\!\overline{u_{n}(y)}e^{-\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)}\frac{(u_{n}(x)\!-\!u_{n}(y)e^{\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)})(\varepsilonta_{R}(x)\!-\!\varepsilonta_{R}(y))}{|x-y|^{N+2s}} dx dy\Bigr)\\
&+\iint_{\mathbb{R}^{2N}} \!\!\!\varepsilonta_{R}(x)\frac{|u_{n}(x)\!-\!u_{n}(y)e^{\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)}|^{2}}{|x-y|^{N+2s}} dx dy,
\varepsilonnd{align*}
and choosing $R>0$ large enough such that $\Lambda_{\varepsilon}\subset B_{\frac{R}{2}}$, we can use $(g_3)$-$(ii)$ and \varepsilonqref{Kbound} to get
\begin{align}\label{PS1}
&\iint_{\mathbb{R}^{2N}} \varepsilonta_{R}(x)\frac{|u_{n}(x)-u_{n}(y)e^{\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)}|^{2}}{|x-y|^{N+2s}}\, dx dy+\int_{\mathbb{R}^{N}} V_{\varepsilon}(x)\varepsilonta_{R} |u_{n}|^{2}\, dx\nonumber\\
&\leq -\mathbb{R}e\Bigl(\iint_{\mathbb{R}^{2N}} \!\!\!\overline{u_{n}(y)}e^{-\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)}\frac{(u_{n}(x)\!-\!u_{n}(y)e^{\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)})(\varepsilonta_{R}(x)\!-\!\varepsilonta_{R}(y))}{|x-y|^{N+2s}} dx dy\Bigr) \nonumber\\
&+\frac{1}{2}\int_{\mathbb{R}^{N}} V_{\varepsilon}(x) |u_{n}|^{2} \varepsilonta_{R} \, dx+o_{n}(1).
\varepsilonnd{align}
From the H\"older inequality and the boundedness of $(u_{n})$ in $H^{s}_{\e}$ it follows that
\begin{align}\label{PS2}
&\Bigl|\mathbb{R}e\Bigl(\iint_{\mathbb{R}^{2N}} \!\!\!\overline{u_{n}(y)}e^{-\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)}\frac{(u_{n}(x)\!-\!u_{n}(y)e^{\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)})(\varepsilonta_{R}(x)\!-\!\varepsilonta_{R}(y))}{|x-y|^{N+2s}} dx dy\Bigr)\Bigr| \nonumber\\
&\leq \Bigl(\iint_{\mathbb{R}^{2N}} \frac{|u_{n}(x)-u_{n}(y)e^{\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)}|^{2}}{|x-y|^{N+2s}}dxdy \Bigr)^{\frac{1}{2}} \times \nonumber\\
&\quad \times \Bigl(\iint_{\mathbb{R}^{2N}} |\overline{u_{n}(y)}|^{2}\frac{|\varepsilonta_{R}(x)-\varepsilonta_{R}(y)|^{2}}{|x-y|^{N+2s}} dxdy\Bigr)^{\frac{1}{2}} \nonumber\\
&\leq C \Bigl(\iint_{\mathbb{R}^{2N}} |u_{n}(y)|^{2}\frac{|\varepsilonta_{R}(x)-\varepsilonta_{R}(y)|^{2}}{|x-y|^{N+2s}} \, dxdy\Bigr)^{\frac{1}{2}}.
\varepsilonnd{align}
By using Lemma 2.1 in \cite{A6} we can see that
\begin{equation}\label{PS3}
\lim_{R\rightarrow \infty}\limsup_{n\rightarrow \infty} \iint_{\mathbb{R}^{2N}} |u_{n}(y)|^{2}\frac{|\varepsilonta_{R}(x)-\varepsilonta_{R}(y)|^{2}}{|x-y|^{N+2s}} \, dxdy=0.
\varepsilonnd{equation}
Then, putting together \varepsilonqref{PS1}, \varepsilonqref{PS2} and \varepsilonqref{PS3} we can deduce that \varepsilonqref{DF} holds true.
\noindent
{\bf Step $2$}: Let us prove that $u_{n}\rightarrow u$ in $H^{s}_{\varepsilon}$ as $n\rightarrow \infty$.\\
Since $u_{n}\rightharpoonup u$ in $H^{s}_{\e}$ and $\langle J'_{\varepsilon}(u_{n}),u_{n}\rangle=\langle J'_{\varepsilon}(u_{n}),u\rangle=o_{n}(1)$
we can note that
$$
\|u_{n}\|^{2}_{\varepsilon}-\|u\|^{2}_{\varepsilon}=\|u_{n}-u\|^{2}_{\varepsilon}+o_{n}(1)=\int_{\mathbb{R}^{N}} \tilde{K}_{\varepsilon}(u_{n})g_{\varepsilon}(x, |u_{n}|^{2})(|u_{n}|^{2}-|u|^{2})dx+o_{n}(1).
$$
Therefore, being $H^{s}_{\e}$ be a Hilbert space, it is enough to show that
$$
\int_{\mathbb{R}^{N}} \tilde{K}_{\varepsilon}(u_{n})g_{\varepsilon}(x, |u_{n}|^{2})(|u_{n}|^{2}-|u|^{2})dx=o_{n}(1).
$$
By Lemma \ref{lemK} we know that $|\tilde{K}_{\varepsilon}(u_{n})|\leq C$ for all $n\in \mathbb{N}$. Since $|u_{n}|\rightarrow |u|$ in $L^{r}(B_{R})$ for all $r\in [2, 2^{*}_{s})$ and $R>0$, we obtain
\begin{align}
\left|\int_{B_{R}} \tilde{K}_{\varepsilon}(u_{n})g_{\varepsilon}(x, |u_{n}|^{2})(|u_{n}|^{2}-|u|^{2})dx\right|\leq C\int_{B_{R}} |g_{\varepsilon}(x, |u_{n}|^{2})(|u_{n}|^{2}-|u|^{2})|dx\rightarrow 0.
\varepsilonnd{align}
By the Step $1$ and Theorem \ref{Sembedding}, for any $\varepsilonta>0$ there exists $R_{\varepsilonta}>0$ such that
$$
\limsup_{n\rightarrow \infty} \int_{\mathbb{R}^{N}\setminus B_{R}} \tilde{K}_{\varepsilon}(u_{n}) |g(\varepsilon x, |u_{n}|^{2})|u_{n}|^{2}| \, dx\leq C\varepsilonta.
$$
In similar way, from H\"older inequality, we can see that
$$
\limsup_{n\rightarrow \infty} \int_{\mathbb{R}^{N}\setminus B_{R}} \tilde{K}_{\varepsilon}(u_{n}) |g(\varepsilon x, |u_{n}|^{2})|u|^{2}| \, dx\leq C\varepsilonta.
$$
Taking into account the above limits we can infer that
$$
\lim_{n\rightarrow \infty} \int_{\mathbb{R}^{N}} \tilde{K}_{\varepsilon}(u_{n}) g(\varepsilon x, |u_{n}|^{2})(|u_{n}|^{2}-|u|^{2}) \, dx=0.
$$
This ends the proof of Lemma \ref{PSc}.
\varepsilonnd{proof}
| 3,262 | 38,947 |
en
|
train
|
0.43.6
|
\section{Concentration of solutions to \varepsilonqref{P}}
In this section we give the proof of the main result of this paper.
Firstly, we consider the limit problem associated with \varepsilonqref{R}, that is
\begin{equation}\label{APe}
(-\Delta)^{s} u + V_{0}u = \left(\frac{1}{|x|^{\mu}}*F(|u|^{2})\right)f(|u|^{2})u \mbox{ in } \mathbb{R}^{N},
\varepsilonnd{equation}
and the corresponding energy functional $J_{0}: H^{s}_{0}\rightarrow \mathbb{R}$ given by
$$
J_{0}(u)=\frac{1}{2}\|u\|^{2}_{V_{0}}-\mathfrak{F}(u),
$$
where $H_{0}^{s}$ is the space $H^{s}(\mathbb{R}^{N}, \mathbb{R})$ endowed with the norm
$$
\|u\|^{2}_{V_{0}}=[u]^{2}+\int_{\mathbb{R}^{N}} V_{0} u^{2}\,dx,
$$
and
$$
\mathfrak{F}(u)=\frac{1}{4}\int_{\mathbb{R}^{N}} \left(\frac{1}{|x|^{\mu}}*F(|u|^{2})\right)F(|u|^{2})\, dx.
$$
As in the previous section, it is easy to see that $J_{0}$ has a mountain pass geometry and we denote by $c_{V_{0}}$ the mountain pass level of the functional $J_{0}$.
Let us introduce the Nehari manifold associated with (\ref{Pe}), that is
\begin{equation*}
\mathcal{N}_{\varepsilon}:= \{u\in H^{s}_{\e} \setminus \{0\} : \langle J_{\varepsilon}'(u), u \rangle =0\},
\varepsilonnd{equation*}
and we denote by $\mathcal{N}_{0}$ the Nehari manifold associated with \varepsilonqref{APe}.
It is standard to verify (see \cite{W}) that $c_{\varepsilon}$ can be characterized as
$$
c_{\varepsilon}=\inf_{u\in H^{s}_{\e}\setminus\{0\}} \sup_{t\geq 0} J_{\varepsilon}(t u)=\inf_{u\in \mathcal{N}_{\varepsilon}} J_{\varepsilon}(u).
$$
In the next result we stress an interesting relation between $c_{\varepsilon}$ and $c_{V_{0}}$.
\begin{lemma}\label{AMlem1}
The numbers $c_{\varepsilon}$ and $c_{V_{0}}$ satisfy the following inequality
$$
\limsup_{\varepsilon\rightarrow 0} c_{\varepsilon}\leq c_{V_{0}}.
$$
\varepsilonnd{lemma}
\begin{proof}
In view of Lemma $3.3$ in \cite{Apota}, there exists a ground state $w\in H^{s}(\mathbb{R}^{N}, \mathbb{R})$ to the autonomous problem \varepsilonqref{APe}, so that $J'_{0}(w)=0$ and $J_{0}(w)=c_{V_{0}}$. Moreover, we know that $w\in C^{0, \mu}(\mathbb{R}^{N})$ and $w>0$ in $\mathbb{R}^{N}$.
In what follows, we show that $w$ satisfies the following useful estimate:
\begin{equation}\label{remdecay}
0<w(x)\leq \frac{C}{|x|^{N+2s}} \mbox{ for large } |x|.
\varepsilonnd{equation}
By using $(f_1)$, $\lim_{|x|\rightarrow\infty}w(x)=0$ and the boundedness of the convolution term (see proof of Lemma \ref{lemK}) we can find $R>0$ such that $\left(\frac{1}{|x|^{\mu}}*F(w^{2})\right)f(w^{2})w\leq \frac{V_{0}}{2}w$ in $B_{R}^{c}$. In particular we have
\begin{equation}\label{BBMP1}
(-\Delta)^{s}w+\frac{V_{0}}{2}w=\left(\frac{1}{|x|^{\mu}}*F(w^{2})\right)f(w^{2})w-\left(V_{0}-\frac{V_{0}}{2}\right)w\leq 0 \mbox{ in } B_{R}^{c}.
\varepsilonnd{equation}
In view of Lemma $4.2$ in \cite{FQT} and by rescaling, we know that there exists a positive function $w_{1}$ and a constant $C_{1}>0$ such that for large $|x|>R$ it holds that $w_{1}(x)=C_{1}|x|^{-(N+2s)}$ and
\begin{equation}\label{BBMP2}
(-\Delta)^{s}w_{1}+\frac{V_{0}}{2}w_{1}\geq 0 \mbox{ in } B^{c}_{R}.
\varepsilonnd{equation}
Taking into account the continuity of $w$ and $w_{1}$ there exists $C_{2}>0$ such that $w_{2}(x)=w(x)-C_{2}w_{1}(x)\leq 0$ on $|x|=R$ (taking $R$ larger if necessary). Moreover, we can see that $(-\Delta)^{s}w_{2}+\frac{V_{0}}{2}w_{2}\leq 0$ for $|x|\geq R$ and by using the maximum principle we can infer that $w_{2}\leq 0$ in $B_{R}^{c}$, that is $w\leq C_{2}w_{1}$ in $B_{R}^{c}$. This fact implies that \varepsilonqref{remdecay} holds true.
Now, fix a cut-off function $\varepsilonta\in C^{\infty}_{c}(\mathbb{R}^{N}, [0,1])$ such that $\varepsilonta=1$ in a neighborhood of zero $B_{\frac{\delta}{2}}$ and $\supp(\varepsilonta)\subset B_{\delta}\subset \Lambda$ for some $\delta>0$.
Let us define $w_{\varepsilon}(x):=\varepsilonta_{\varepsilon}(x)w(x) e^{\imath A(0)\cdot x}$, with $\varepsilonta_{\varepsilon}(x)=\varepsilonta(\varepsilon x)$ for $\varepsilon>0$, and we observe that $|w_{\varepsilon}|=\varepsilonta_{\varepsilon}w$ and $w_{\varepsilon}\in H^{s}_{\e}$ in view of Lemma \ref{aux}. Let us prove that
\begin{equation}\label{limwr}
\lim_{\varepsilon\rightarrow 0}\|w_{\varepsilon}\|^{2}_{\varepsilon}=\|w\|_{0}^{2}\in(0, \infty).
\varepsilonnd{equation}
Clearly, $\int_{\mathbb{R}^{N}} V_{\varepsilon}(x)|w_{\varepsilon}|^{2}dx\rightarrow \int_{\mathbb{R}^{N}} V_{0} |w|^{2}dx$. Now, we show that
\begin{equation}\label{limwr*}
\lim_{\varepsilon\rightarrow 0}[w_{\varepsilon}]^{2}_{A_{\varepsilon}}=[w]^{2}.
\varepsilonnd{equation}
We note that, in view of Lemma $5$ in \cite{PP}, we have
\begin{equation}\label{PPlem}
[\varepsilonta_{\varepsilon} w]\rightarrow [w] \mbox{ as } \varepsilon\rightarrow 0.
\varepsilonnd{equation}
On the other hand
\begin{align*}
&[w_{\varepsilon}]_{A_{\varepsilon}}^{2}\nonumber \\
&=\iint_{\mathbb{R}^{2N}} \frac{|e^{\imath A(0)\cdot x}\varepsilonta_{\varepsilon}(x)w(x)-e^{\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)}e^{\imath A(0)\cdot y} \varepsilonta_{\varepsilon}(y)w(y)|^{2}}{|x-y|^{N+2s}} dx dy \nonumber \\
&=[\varepsilonta_{\varepsilon} w]^{2}
+\iint_{\mathbb{R}^{2N}} \frac{\varepsilonta_{\varepsilon}^2(y)w^2(y) |e^{\imath [A_{\varepsilon}(\frac{x+y}{2})-A(0)]\cdot (x-y)}-1|^{2}}{|x-y|^{N+2s}} dx dy\\
&+2\mathbb{R}e \iint_{\mathbb{R}^{2N}} \frac{(\varepsilonta_{\varepsilon}(x)w(x)-\varepsilonta_{\varepsilon}(y)w(y))\varepsilonta_{\varepsilon}(y)w(y)(1-e^{-\imath [A_{\varepsilon}(\frac{x+y}{2})-A(0)]\cdot (x-y)})}{|x-y|^{N+2s}} dx dy \\
&=: [\varepsilonta_{\varepsilon} w]^{2}+X_{\varepsilon}+2Y_{\varepsilon}.
\varepsilonnd{align*}
Taking into account
$|Y_{\varepsilon}|\leq [\varepsilonta_{\varepsilon} w] \sqrt{X_{\varepsilon}}$ and \varepsilonqref{PPlem}, we need to prove that $X_{\varepsilon}\rightarrow 0$ as $\varepsilon\rightarrow 0$ to deduce that \varepsilonqref{limwr*} holds true.
| 2,236 | 38,947 |
en
|
train
|
0.43.7
|
Now, fix a cut-off function $\varepsilonta\in C^{\infty}_{c}(\mathbb{R}^{N}, [0,1])$ such that $\varepsilonta=1$ in a neighborhood of zero $B_{\frac{\delta}{2}}$ and $\supp(\varepsilonta)\subset B_{\delta}\subset \Lambda$ for some $\delta>0$.
Let us define $w_{\varepsilon}(x):=\varepsilonta_{\varepsilon}(x)w(x) e^{\imath A(0)\cdot x}$, with $\varepsilonta_{\varepsilon}(x)=\varepsilonta(\varepsilon x)$ for $\varepsilon>0$, and we observe that $|w_{\varepsilon}|=\varepsilonta_{\varepsilon}w$ and $w_{\varepsilon}\in H^{s}_{\e}$ in view of Lemma \ref{aux}. Let us prove that
\begin{equation}\label{limwr}
\lim_{\varepsilon\rightarrow 0}\|w_{\varepsilon}\|^{2}_{\varepsilon}=\|w\|_{0}^{2}\in(0, \infty).
\varepsilonnd{equation}
Clearly, $\int_{\mathbb{R}^{N}} V_{\varepsilon}(x)|w_{\varepsilon}|^{2}dx\rightarrow \int_{\mathbb{R}^{N}} V_{0} |w|^{2}dx$. Now, we show that
\begin{equation}\label{limwr*}
\lim_{\varepsilon\rightarrow 0}[w_{\varepsilon}]^{2}_{A_{\varepsilon}}=[w]^{2}.
\varepsilonnd{equation}
We note that, in view of Lemma $5$ in \cite{PP}, we have
\begin{equation}\label{PPlem}
[\varepsilonta_{\varepsilon} w]\rightarrow [w] \mbox{ as } \varepsilon\rightarrow 0.
\varepsilonnd{equation}
On the other hand
\begin{align*}
&[w_{\varepsilon}]_{A_{\varepsilon}}^{2}\nonumber \\
&=\iint_{\mathbb{R}^{2N}} \frac{|e^{\imath A(0)\cdot x}\varepsilonta_{\varepsilon}(x)w(x)-e^{\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)}e^{\imath A(0)\cdot y} \varepsilonta_{\varepsilon}(y)w(y)|^{2}}{|x-y|^{N+2s}} dx dy \nonumber \\
&=[\varepsilonta_{\varepsilon} w]^{2}
+\iint_{\mathbb{R}^{2N}} \frac{\varepsilonta_{\varepsilon}^2(y)w^2(y) |e^{\imath [A_{\varepsilon}(\frac{x+y}{2})-A(0)]\cdot (x-y)}-1|^{2}}{|x-y|^{N+2s}} dx dy\\
&+2\mathbb{R}e \iint_{\mathbb{R}^{2N}} \frac{(\varepsilonta_{\varepsilon}(x)w(x)-\varepsilonta_{\varepsilon}(y)w(y))\varepsilonta_{\varepsilon}(y)w(y)(1-e^{-\imath [A_{\varepsilon}(\frac{x+y}{2})-A(0)]\cdot (x-y)})}{|x-y|^{N+2s}} dx dy \\
&=: [\varepsilonta_{\varepsilon} w]^{2}+X_{\varepsilon}+2Y_{\varepsilon}.
\varepsilonnd{align*}
Taking into account
$|Y_{\varepsilon}|\leq [\varepsilonta_{\varepsilon} w] \sqrt{X_{\varepsilon}}$ and \varepsilonqref{PPlem}, we need to prove that $X_{\varepsilon}\rightarrow 0$ as $\varepsilon\rightarrow 0$ to deduce that \varepsilonqref{limwr*} holds true.
Let us observe that for $0<\beta<\alpha/({1+\alpha-s})$ we get
\begin{equation}\label{Ye}
\begin{split}
X_{\varepsilon}
&\leq \int_{\mathbb{R}^{N}} w^{2}(y) dy \int_{|x-y|\geq\varepsilon^{-\beta}} \frac{|e^{\imath [A_{\varepsilon}(\frac{x+y}{2})-A(0)]\cdot (x-y)}-1|^{2}}{|x-y|^{N+2s}} dx\\
&+\int_{\mathbb{R}^{N}} w^{2}(y) dy \int_{|x-y|<\varepsilon^{-\beta}} \frac{|e^{\imath [A_{\varepsilon}(\frac{x+y}{2})-A(0)]\cdot (x-y)}-1|^{2}}{|x-y|^{N+2s}} dx \\
&=:X^{1}_{\varepsilon}+X^{2}_{\varepsilon}.
\varepsilonnd{split}
\varepsilonnd{equation}
Since $|e^{\imath t}-1|^{2}\leq 4$ and $w\in H^{s}(\mathbb{R}^{N}, \mathbb{R})$, we can see that
\begin{equation}\label{Ye1}
X_{\varepsilon}^{1}\leq C \int_{\mathbb{R}^{N}} w^{2}(y) dy \int_{\varepsilon^{-\beta}}^\infty \rho^{-1-2s} d\rho\leq C \varepsilon^{2\beta s} \rightarrow 0.
\varepsilonnd{equation}
Now, by using $|e^{\imath t}-1|^{2}\leq t^{2}$ for all $t\in \mathbb{R}$, $A\in C^{0,\alpha}(\mathbb{R}^N,\mathbb{R}^N)$ for $\alpha\in(0,1]$, and $|x+y|^{2}\leq 2(|x-y|^{2}+4|y|^{2})$, we obtain
\begin{equation}\label{Ye2}
\begin{split}
X^{2}_{\varepsilon}&
\leq \int_{\mathbb{R}^{N}} w^{2}(y) dy \int_{|x-y|<\varepsilon^{-\beta}} \frac{|A_{\varepsilon}\left(\frac{x+y}{2}\right)-A(0)|^{2} }{|x-y|^{N+2s-2}} dx \\
&\leq C\varepsilon^{2\alpha} \int_{\mathbb{R}^{N}} w^{2}(y) dy \int_{|x-y|<\varepsilon^{-\beta}} \frac{|x+y|^{2\alpha} }{|x-y|^{N+2s-2}} dx \\
&\leq C\varepsilon^{2\alpha} \left(\int_{\mathbb{R}^{N}} w^{2}(y) dy \int_{|x-y|<\varepsilon^{-\beta}} \frac{1 }{|x-y|^{N+2s-2-2\alpha}} dx\right.\\
&\qquad\qquad+ \left. \int_{\mathbb{R}^{N}} |y|^{2\alpha} w^{2}(y) dy \int_{|x-y|<\varepsilon^{-\beta}} \frac{1}{|x-y|^{N+2s-2}} dx\right) \\
&=: C\varepsilon^{2\alpha} (X^{2, 1}_{\varepsilon}+X^{2, 2}_{\varepsilon}).
\varepsilonnd{split}
\varepsilonnd{equation}
Then
\begin{equation}\label{Ye21}
X^{2, 1}_{\varepsilon}
= C \int_{\mathbb{R}^{N}} w^{2}(y) dy \int_0^{\varepsilon^{-\beta}} \rho^{1+2\alpha-2s} d\rho
\leq C\varepsilon^{-2\beta(1+\alpha-s)}.
\varepsilonnd{equation}
On the other hand, using \varepsilonqref{remdecay}, we can infer that
\begin{equation}\label{Ye22}
\begin{split}
X^{2, 2}_{\varepsilon}
&\leq C \int_{\mathbb{R}^{N}} |y|^{2\alpha} w^{2}(y) dy \int_0^{\varepsilon^{-\beta}}\rho^{1-2s} d\rho \\
&\leq C \varepsilon^{-2\beta(1-s)} \left[\int_{B_1(0)} w^{2}(y) dy + \int_{B_1^c(0)} \frac{1}{|y|^{2(N+2s)-2\alpha}} dy \right] \\
&\leq C \varepsilon^{-2\beta(1-s)}.
\varepsilonnd{split}
\varepsilonnd{equation}
Taking into account \varepsilonqref{Ye}, \varepsilonqref{Ye1}, \varepsilonqref{Ye2}, \varepsilonqref{Ye21} and \varepsilonqref{Ye22} we have $X_{\varepsilon}\rightarrow 0$, and then \varepsilonqref{limwr} holds.
Now, let $t_{\varepsilon}>0$ be the unique number such that
\begin{equation*}
J_{\varepsilon}(t_{\varepsilon} w_{\varepsilon})=\max_{t\geq 0} J_{\varepsilon}(t w_{\varepsilon}).
\varepsilonnd{equation*}
As a consequence, $t_{\varepsilon}$ satisfies
\begin{align}\label{AS1}
\|w_{\varepsilon}\|_{\varepsilon}^{2}&=\int_{\mathbb{R}^{N}} \left(\frac{1}{|x|^{\mu}}*G(\varepsilon x, t_{\varepsilon}^{2} |w_{\varepsilon}|^{2})\right)g(\varepsilon x, t_{\varepsilon}^{2} |w_{\varepsilon}|^{2}) |w_{\varepsilon}|^{2}dx \nonumber \\
&=\int_{\mathbb{R}^{N}} \left(\frac{1}{|x|^{\mu}}*F(t_{\varepsilon}^{2} |w_{\varepsilon}|^{2})\right) f(t_{\varepsilon}^{2} |w_{\varepsilon}|^{2}) |w_{\varepsilon}|^{2}dx
\varepsilonnd{align}
where we used $supp(\varepsilonta)\subset \Lambda$ and $g=f$ on $\Lambda$.\\
Let us prove that $t_{\varepsilon}\rightarrow 1$ as $\varepsilon\rightarrow 0$. Since $\varepsilonta=1$ in $B_{\frac{\delta}{2}}$, $w$ is a continuous positive function, and recalling that $f(t)$ and $F(t)/t$ are both increasing, we have
$$
\|w_{\varepsilon}\|_{\varepsilon}^{2}\geq \frac{F(t_{\varepsilon}^{2}\alpha_{0}^{2})}{\alpha_{0}^{2}} f(t_{\varepsilon}^{2}\alpha^{2}_{0})\int_{B_{\frac{\delta}{2}}}\int_{B_{\frac{\delta}{2}}}
\frac{|w(x)|^{2} |w(y)|^{2}}{|x-y|^{\mu}}dx dy,
$$
where $\alpha_{0}=\min_{\bar{B}_{\frac{\delta}{2}}} w>0$. \\
Let us prove that $t_{\varepsilon}\rightarrow t_{0}\in (0, \infty)$ as $\varepsilon\rightarrow 0$. Indeed, if $t_{\varepsilon}\rightarrow \infty$ as $\varepsilon\rightarrow 0$ then we can use $(f_3)$ to deduce that $\|w\|_{0}^{2}= \infty$ which gives a contradiction due to \varepsilonqref{limwr}.
When $t_{\varepsilon}\rightarrow 0$ as $\varepsilon\rightarrow 0$ we can use $(f_1)$ to infer that $\|w\|_{0}^{2}= 0$ which is impossible in view of \varepsilonqref{limwr}.\\
Then, taking the limit as $\varepsilon\rightarrow 0$ in \varepsilonqref{AS1} and using \varepsilonqref{limwr}, we can deduce that
\begin{equation}\label{AS2}
\|w\|_{0}^{2}=\int_{\mathbb{R}^{N}} \left(\frac{1}{|x|^{\mu}}*F(t_{0}^{2}|w|^{2})\right)f(t_{0}^{2} |w|^{2}) |w|^{2}dx.
\varepsilonnd{equation}
Since $w\in \mathcal{N}_{0}$ and using $(f_3)$, we obtain $t_{0}=1$. Hence, from the Dominated Convergence Theorem, we can see that $\lim_{\varepsilon\rightarrow 0} J_{\varepsilon}(t_{\varepsilon} w_{\varepsilon})=J_{0}(w)=c_{V_{0}}$.
Recalling that $c_{\varepsilon}\leq \max_{t\geq 0} J_{\varepsilon}(t w_{\varepsilon})=J_{\varepsilon}(t_{\varepsilon} w_{\varepsilon})$, we can infer that
$\limsup_{\varepsilon\rightarrow 0} c_{\varepsilon}\leq c_{V_{0}}$.
\varepsilonnd{proof}
\noindent
Arguing as in \cite{Apota}, we can deduce the following result for the autonomous problem:
\begin{lemma}\label{FS}
Let $(u_{n})\subset \mathcal{N}_{0}$ be a sequence satisfying $J_{0}(u_{n})\rightarrow c_{V_{0}}$. Then, up to subsequences, the following alternatives holds:
\begin{enumerate}
\item [$(i)$] $(u_{n})$ strongly converges in $H^{s}(\mathbb{R}^{N}, \mathbb{R})$,
\item [$(ii)$] there exists a sequence $(\tilde{y}_{n})\subset \mathbb{R}^{N}$ such that, up to a subsequence, $v_{n}(x)=u_{n}(x+\tilde{y}_{n})$ converges strongly in $H^{s}(\mathbb{R}^{N}, \mathbb{R})$.
\varepsilonnd{enumerate}
In particular, there exists a minimizer for $c_{V_{0}}$.
\varepsilonnd{lemma}
| 3,226 | 38,947 |
en
|
train
|
0.43.8
|
\noindent
Arguing as in \cite{Apota}, we can deduce the following result for the autonomous problem:
\begin{lemma}\label{FS}
Let $(u_{n})\subset \mathcal{N}_{0}$ be a sequence satisfying $J_{0}(u_{n})\rightarrow c_{V_{0}}$. Then, up to subsequences, the following alternatives holds:
\begin{enumerate}
\item [$(i)$] $(u_{n})$ strongly converges in $H^{s}(\mathbb{R}^{N}, \mathbb{R})$,
\item [$(ii)$] there exists a sequence $(\tilde{y}_{n})\subset \mathbb{R}^{N}$ such that, up to a subsequence, $v_{n}(x)=u_{n}(x+\tilde{y}_{n})$ converges strongly in $H^{s}(\mathbb{R}^{N}, \mathbb{R})$.
\varepsilonnd{enumerate}
In particular, there exists a minimizer for $c_{V_{0}}$.
\varepsilonnd{lemma}
\noindent
Now, we prove the following useful compactness result.
\begin{lemma}\label{prop3.3}
Let $\varepsilon_{n}\rightarrow 0$ and $(u_{n})\subset H^{s}_{\varepsilon_{n}}$ such that $J_{\varepsilon_{n}}(u_{n})=c_{\varepsilon_{n}}$ and $J'_{\varepsilon_{n}}(u_{n})=0$. Then there exists $(\tilde{y}_{n})\subset \mathbb{R}^{N}$ such that $v_{n}(x)=|u_{n}|(x+\tilde{y}_{n})$ has a convergent subsequence in $H^{s}(\mathbb{R}^{N}, \mathbb{R})$. Moreover, up to a subsequence, $y_{n}=\varepsilon_{n} \tilde{y}_{n}\rightarrow y_{0}$ for some $y_{0}\in \Lambda$ such that $V(y_{0})=V_{0}$.
\varepsilonnd{lemma}
\begin{proof}
Taking into account $\langle J'_{\varepsilon_{n}}(u_{n}), u_{n}\rangle=0$, $J_{\varepsilon_{n}}(u_{n})= c_{\varepsilon_{n}}$, Lemma \ref{AMlem1} and arguing as in Lemma \ref{PSc}, it is easy to see that $(u_{n})$ is bounded in $H^{s}_{\varepsilon_{n}}$ and $\|u_{n}\|^{2}_{\varepsilon_{n}}\leq 4(\kappa+1)$ for all $n\in \mathbb{N}$. Moreover, from Lemma \ref{DI}, we also know that $(|u_{n}|)$ is bounded in $H^{s}(\mathbb{R}^{N}, \mathbb{R})$.\\
Let us prove that there exist a sequence $(\tilde{y}_{n})\subset \mathbb{R}^{N}$ and constants $R>0$ and $\gamma>0$ such that
\begin{equation}\label{sacchi}
\liminf_{n\rightarrow \infty}\int_{B_{R}(\tilde{y}_{n})} |u_{n}|^{2} \, dx\geq \gamma>0.
\varepsilonnd{equation}
Otherwise, if \varepsilonqref{sacchi} does not hold, then for all $R>0$ we have
$$
\lim_{n\rightarrow \infty}\sup_{y\in \mathbb{R}^{N}}\int_{B_{R}(y)} |u_{n}|^{2} \, dx=0.
$$
From the boundedness $(|u_{n}|)$ and Lemma $2.2$ in \cite{FQT} we can see that $|u_{n}|\rightarrow 0$ in $L^{q}(\mathbb{R}^{N}, \mathbb{R})$ for any $q\in (2, 2^{*}_{s})$.
By using $(g_1)$-$(g_2)$ and Lemma \ref{lemK} we can deduce that
\begin{align}\label{glimiti}
\lim_{n\rightarrow \infty}\int_{\mathbb{R}^{N}} \tilde{K}_{\varepsilon_{n}}(u_{n}) g(\varepsilon_{n} x, |u_{n}|^{2}) |u_{n}|^{2} \,dx=0= \lim_{n\rightarrow \infty}\int_{\mathbb{R}^{N}} \tilde{K}_{\varepsilon_{n}}(u_{n}) G(\varepsilon_{n} x, |u_{n}|^{2}) \, dx.
\varepsilonnd{align}
Since $\langle J'_{\varepsilon_{n}}(u_{n}), u_{n}\rangle=0$, we can use \varepsilonqref{glimiti} to deduce that $\|u_{n}\|_{\varepsilon_{n}}\rightarrow 0$ as $n\rightarrow \infty$. This gives a contradiction because $u_{n}\in \mathcal{N}_{\varepsilon_{n}}$ and by using $(g_1)$, $(g_2)$ and Lemma \ref{lemK} we can find $\alpha_{0}>0$ such that $\|u_{n}\|^{2}_{\varepsilon_{n}}\geq \alpha_{0}$ for all $n\in \mathbb{N}$.\\
Set $v_{n}(x)=|u_{n}|(x+\tilde{y}_{n})$. Then $(v_{n})$ is bounded in $H^{s}(\mathbb{R}^{N}, \mathbb{R})$, and we may assume that
$v_{n}\rightharpoonup v\not\varepsilonquiv 0$ in $H^{s}(\mathbb{R}^{N}, \mathbb{R})$ as $n\rightarrow \infty$.
Fix $t_{n}>0$ such that $\tilde{v}_{n}=t_{n} v_{n}\in \mathcal{N}_{0}$. By using Lemma \ref{DI}, we can see that
$$
c_{V_{0}}\leq J_{0}(\tilde{v}_{n})\leq \max_{t\geq 0}J_{\varepsilon_{n}}(tv_{n})= J_{\varepsilon_{n}}(u_{n})=c_{\varepsilon_{n}}
$$
which together with Lemma \ref{AMlem1} implies that $J_{0}(\tilde{v}_{n})\rightarrow c_{V_{0}}$. In particular, $\tilde{v}_{n}\nrightarrow 0$ in $H^{s}(\mathbb{R}^{N}, \mathbb{R})$.
Since $(v_{n})$ and $(\tilde{v}_{n})$ are bounded in $H^{s}(\mathbb{R}^{N}, \mathbb{R})$ and $\tilde{v}_{n}\nrightarrow 0$ in $H^{s}(\mathbb{R}^{N}, \mathbb{R})$, we obtain that $t_{n}\rightarrow t^{*}> 0$.
From the uniqueness of the weak limit, we can deduce that $\tilde{v}_{n}\rightharpoonup \tilde{v}=t^{*}v\not\varepsilonquiv 0$ in $H^{s}(\mathbb{R}^{N}, \mathbb{R})$.
This together with Lemma \ref{FS} gives
\begin{equation}\label{elena}
\tilde{v}_{n}\rightarrow \tilde{v} \mbox{ in } H^{s}(\mathbb{R}^{N}, \mathbb{R}),
\varepsilonnd{equation}
and as a consequence $v_{n}\rightarrow v$ in $H^{s}(\mathbb{R}^{N}, \mathbb{R})$ as $n\rightarrow \infty$.
Now, we set $y_{n}=\varepsilon_{n}\tilde{y}_{n}$. We aim to prove that $(y_{n})$ admits a subsequence, still denoted by $y_{n}$, such that $y_{n}\rightarrow y_{0}$ for some $y_{0}\in \Lambda$ such that $V(y_{0})=V_{0}$. Firstly, we prove that $(y_{n})$ is bounded. Assume by contradiction that, up to a subsequence, $|y_{n}|\rightarrow \infty$ as $n\rightarrow \infty$. Take $R>0$ such that $\Lambda \subset B_{R}(0)$. Since we may suppose that $|y_{n}|>2R$, we have that $|\varepsilon_{n}z+y_{n}|\geq |y_{n}|-|\varepsilon_{n}z|>R$ for any $z\in B_{R/\varepsilon_{n}}$.
Taking into account $(u_{n})\subset \mathcal{N}_{\varepsilon_{n}}$, $(V_{1})$, Lemma \ref{DI} and the change of variable $x\mapsto z+\tilde{y}_{n}$ we get
\begin{align*}
&[v_{n}]^{2}+\int_{\mathbb{R}^{N}} V_{0} v_{n}^{2}\, dx \\
&\leq C_{0}\int_{\mathbb{R}^{N}} g(\varepsilon_{n} z+y_{n}, |v_{n}|^{2}) |v_{n}|^{2} \, dz \\
&\leq C_{0}\int_{B_{\frac{R}{\varepsilon_{n}}}(0)} \tilde{f}(|v_{n}|^{2}) |v_{n}|^{2} \, dz+C_{0}\int_{\mathbb{R}^{N}\setminus B_{\frac{R}{\varepsilon_{n}}}(0)} f(|v_{n}|^{2}) |v_{n}|^{2} \, dz,
\varepsilonnd{align*}
where we used $u_{n}\in \B$ for all $n$ big enough and Lemma \ref{lemK}.
By using $v_{n}\rightarrow v$ in $H^{s}(\mathbb{R}^{N}, \mathbb{R})$ as $n\rightarrow \infty$ and $\tilde{f}(t)\leq \frac{V_{0}}{\varepsilonll_{0}}$ we obtain
$$
\min\left\{1, \frac{V_{0}}{2} \right\} \left([v_{n}]^{2}+\int_{\mathbb{R}^{N}} |v_{n}|^{2}\, dx\right)=o_{n}(1).
$$
Then $v_{n}\rightarrow 0$ in $H^{s}(\mathbb{R}^{N}, \mathbb{R})$ and this is impossible. Therefore, $(y_{n})$ is bounded and we may assume that $y_{n}\rightarrow y_{0}\in \mathbb{R}^{N}$. If $y_{0}\notin \overline{\Lambda}$, we can argue as before to deduce that $v_{n}\rightarrow 0$ in $H^{s}(\mathbb{R}^{N}, \mathbb{R})$, which gives a contradiction. Therefore $y_{0}\in \overline{\Lambda}$, and in view of $(V_2)$, it is enough to verify that $V(y_{0})=V_{0}$ to conclude the proof of lemma. Assume by contradiction that $V(y_{0})>V_{0}$.
Then, by using (\ref{elena}), Fatou's Lemma, the invariance of $\mathbb{R}^{N}$ by translations, Lemma \ref{DI} and Lemma \ref{AMlem1}, we get
\begin{align*}
&c_{V_{0}}=J_{0}(\tilde{v})<\frac{1}{2}[\tilde{v}]^{2}+\frac{1}{2}\int_{\mathbb{R}^{N}} V(y_{0})\tilde{v}^{2} \, dx-\mathfrak{F}(\tilde{v}) \\
&\leq \liminf_{n\rightarrow \infty}\Bigl[\frac{1}{2}[\tilde{v}_{n}]^{2}+\frac{1}{2}\int_{\mathbb{R}^{N}} V(\varepsilon_{n}z+y_{n}) |\tilde{v}_{n}|^{2} \, dz-\mathfrak{F}(\tilde{v}_{n}) \Bigr] \\
&\leq \liminf_{n\rightarrow \infty}\Bigl[\frac{t_{n}^{2}}{2}[|u_{n}|]^{2}+\frac{t_{n}^{2}}{2}\int_{\mathbb{R}^{N}} V(\varepsilon_{n}z) |u_{n}|^{2} \, dz-\mathfrak{F}(t_{n} u_{n}) \Bigr] \\
&\leq \liminf_{n\rightarrow \infty} J_{\varepsilon_{n}}(t_{n} u_{n}) \leq \liminf_{n\rightarrow \infty} J_{\varepsilon_{n}}(u_{n})\leq c_{V_{0}}
\varepsilonnd{align*}
which gives a contradiction.
\varepsilonnd{proof}
| 2,839 | 38,947 |
en
|
train
|
0.43.9
|
\noindent
The next lemma will be fundamental to prove that the solutions of \varepsilonqref{Pe} are also solutions of the original problem \varepsilonqref{P}. We will use a suitable variant of the Moser iteration argument \cite{Moser}.
\begin{lemma}\label{moser}
Let $\varepsilon_{n}\rightarrow 0$ and $u_{n}\in H^{s}_{\varepsilon_{n}}$ be a solution to \varepsilonqref{Pe}.
Then $v_{n}=|u_{n}|(\cdot+\tilde{y}_{n})$ satisfies $v_{n}\in L^{\infty}(\mathbb{R}^{N},\mathbb{R})$ and there exists $C>0$ such that
$$
\|v_{n}\|_{L^{\infty}(\mathbb{R}^{N})}\leq C \mbox{ for all } n\in \mathbb{N},
$$
where $\tilde{y}_{n}$ is given by Lemma \ref{prop3.3}.
Moreover
$$
\lim_{|x|\rightarrow \infty} v_{n}(x)=0 \mbox{ uniformly in } n\in \mathbb{N}.
$$
\varepsilonnd{lemma}
\begin{proof}
For any $L>0$ we define $u_{L,n}:=\min\{|u_{n}|, L\}\geq 0$ and we set $v_{L, n}=u_{L,n}^{2(\beta-1)}u_{n}$ where $\beta>1$ will be chosen later.
Taking $v_{L, n}$ as test function in (\ref{Pe}) we can see that
\begin{align}\label{conto1FF}
&\mathbb{R}e\Bigl(\iint_{\mathbb{R}^{2N}} \frac{(u_{n}(x)-u_{n}(y)e^{\imath A(\frac{x+y}{2})\cdot (x-y)})}{|x-y|^{N+2s}} \times \nonumber\\
&\quad \times \overline{((u_{n}u_{L,n}^{2(\beta-1)})(x)-(u_{n}u_{L,n}^{2(\beta-1)})(y)e^{\imath A(\frac{x+y}{2})\cdot (x-y)})} dx dy\Bigr) \nonumber \\
&=\int_{\mathbb{R}^{N}} \tilde{K}_{\varepsilon}(u_{n}) g(\varepsilon_{n} x, |u_{n}|^{2}) |u_{n}|^{2}u_{L,n}^{2(\beta-1)} \,dx-\int_{\mathbb{R}^{N}} V(\varepsilon_{n} x) |u_{n}|^{2} u_{L,n}^{2(\beta-1)} \, dx.
\varepsilonnd{align}
Let us observe that
\begin{align*}
&\mathbb{R}e\left[(u_{n}(x)-u_{n}(y)e^{\imath A(\frac{x+y}{2})\cdot (x-y)})\overline{(u_{n}u_{L,n}^{2(\beta-1)}(x)-u_{n}u_{L,n}^{2(\beta-1)}(y)e^{\imath A(\frac{x+y}{2})\cdot (x-y)})}\right] \\
&=\mathbb{R}e\Bigl[|u_{n}(x)|^{2}v_{L}^{2(\beta-1)}(x)-u_{n}(x)\overline{u_{n}(y)} u_{L,n}^{2(\beta-1)}(y)e^{-\imath A(\frac{x+y}{2})\cdot (x-y)}\\
&\quad-u_{n}(y)\overline{u_{n}(x)} u_{L,n}^{2(\beta-1)}(x) e^{\imath A(\frac{x+y}{2})\cdot (x-y)} +|u_{n}(y)|^{2}u_{L,n}^{2(\beta-1)}(y) \Bigr] \\
&\geq (|u_{n}(x)|^{2}u_{L,n}^{2(\beta-1)}(x)-|u_{n}(x)||u_{n}(y)|u_{L,n}^{2(\beta-1)}(y) \\
&\quad -|u_{n}(y)||u_{n}(x)|u_{L,n}^{2(\beta-1)}(x)+|u_{n}(y)|^{2}u^{2(\beta-1)}_{L,n}(y) \\
&=(|u_{n}(x)|-|u_{n}(y)|)(|u_{n}(x)|u_{L,n}^{2(\beta-1)}(x)-|u_{n}(y)|u_{L,n}^{2(\beta-1)}(y)),
\varepsilonnd{align*}
which implies that
\begin{align}\label{realeF}
&\mathbb{R}e\Bigl(\iint_{\mathbb{R}^{2N}} \frac{(u_{n}(x)-u_{n}(y)e^{\imath A(\frac{x+y}{2})\cdot (x-y)})}{|x-y|^{N+2s}} \times \nonumber\\
&\quad \times \overline{((u_{n}u_{L,n}^{2(\beta-1)})(x)-(u_{n}u_{L,n}^{2(\beta-1)})(y)e^{\imath A(\frac{x+y}{2})\cdot (x-y)})} dx dy\Bigr) \nonumber\\
&\geq \iint_{\mathbb{R}^{2N}} \frac{(|u_{n}(x)|-|u_{n}(y)|)}{|x-y|^{N+2s}} (|u_{n}(x)|u_{L,n}^{2(\beta-1)}(x)-|u_{n}(y)|u_{L,n}^{2(\beta-1)}(y))\, dx dy.
\varepsilonnd{align}
As in \cite{Apota}, for all $t\geq 0$, we define
\begin{equation*}
\gamma(t)=\gamma_{L, \beta}(t)=t t_{L}^{2(\beta-1)}
\varepsilonnd{equation*}
where $t_{L}=\min\{t, L\}$.
Since $\gamma$ is an increasing function we have
\begin{align*}
(a-b)(\gamma(a)- \gamma(b))\geq 0 \quad \mbox{ for any } a, b\in \mathbb{R}.
\varepsilonnd{align*}
Let
\begin{equation*}
\Lambda(t)=\frac{|t|^{2}}{2} \quad \mbox{ and } \quad \Gamma(t)=\int_{0}^{t} (\gamma'(\tau))^{\frac{1}{2}} d\tau.
\varepsilonnd{equation*}
Since
\begin{equation}\label{Gg}
\Lambda'(a-b)(\gamma(a)-\gamma(b))\geq |\Gamma(a)-\Gamma(b)|^{2} \mbox{ for any } a, b\in\mathbb{R},
\varepsilonnd{equation}
we get
\begin{align}\label{Gg1}
|\Gamma(|u_{n}(x)|)- \Gamma(|u_{n}(y)|)|^{2} \leq (|u_{n}(x)|- |u_{n}(y)|)((|u_{n}|u_{L,n}^{2(\beta-1)})(x)- (|u_{n}|u_{L,n}^{2(\beta-1)})(y)).
\varepsilonnd{align}
Putting together \varepsilonqref{realeF} and \varepsilonqref{Gg1}, we can see that
\begin{align}\label{conto1FFF}
&\mathbb{R}e\Bigl(\iint_{\mathbb{R}^{2N}} \frac{(u_{n}(x)-u_{n}(y)e^{\imath A(\frac{x+y}{2})\cdot (x-y)})}{|x-y|^{N+2s}} \times \nonumber\\
&\quad \times \overline{(u_{n}u_{L,n}^{2(\beta-1)}(x)\!-\!u_{n}u_{L,n}^{2(\beta-1)}(y)e^{\imath A(\frac{x+y}{2})\cdot (x-y)})} dx dy\Bigr) \nonumber\\
&\geq [\Gamma(|u_{n}|)]^{2}.
\varepsilonnd{align}
Since $\Gamma(|u_{n}|)\geq \frac{1}{\beta} |u_{n}| u_{L,n}^{\beta-1}$ and using the fractional Sobolev embedding $\mathcal{D}^{s,2}(\mathbb{R}^{N}, \mathbb{R})\subset L^{2^{*}_{s}}(\mathbb{R}^{N}, \mathbb{R})$ (see \cite{DPV}), we can infer that
\begin{equation}\label{SS1}
[\Gamma(|u_{n}|)]^{2}\geq S_{*} \|\Gamma(|u_{n}|)\|^{2}_{L^{2^{*}_{s}}(\mathbb{R}^{N})}\geq \left(\frac{1}{\beta}\right)^{2} S_{*}\||u_{n}| u_{L,n}^{\beta-1}\|^{2}_{L^{2^{*}_{s}}(\mathbb{R}^{N})}.
\varepsilonnd{equation}
Then \varepsilonqref{conto1FF}, \varepsilonqref{conto1FFF} and \varepsilonqref{SS1} yield
\begin{align}\label{BMS}
&\left(\frac{1}{\beta}\right)^{2} S_{*}\||u_{n}| u_{L,n}^{\beta-1}\|^{2}_{L^{2^{*}_{s}}(\mathbb{R}^{N})}+\int_{\mathbb{R}^{N}} V(\varepsilon_{n} x)|u_{n}|^{2}u_{L,n}^{2(\beta-1)} dx\nonumber\\
&\leq \int_{\mathbb{R}^{N}} \tilde{K}_{\varepsilon_{n}}(u_{n}) g(\varepsilon_{n}x, |u_{n}|^{2}) |u_{n}|^{2} u_{L,n}^{2(\beta-1)} dx.
\varepsilonnd{align}
By $(g_1)$ and $(g_2)$, we know that for any $\xi>0$ there exists $C_{\xi}>0$ such that
\begin{equation}\label{SS2}
g(x, t^{2})t^{2}\leq \xi |t|^{2}+C_{\xi}|t|^{2^{*}_{s}} \mbox{ for any } (x, t)\in \mathbb{R}^{N}\times\mathbb{R}.
\varepsilonnd{equation}
Hence, using \varepsilonqref{BMS}, \varepsilonqref{SS2}, $u_{n}\in \B$, Lemma \ref{lemK} and choosing $\xi>0$ sufficiently small, we can see that
\begin{equation}\label{simo1}
\|w_{L,n}\|_{L^{2^{*}_{s}}(\mathbb{R}^{N})}^{2}\leq C \beta^{2} \int_{\mathbb{R}^{N}} |u_{n}|^{q}u_{L,n}^{2(\beta-1)},
\varepsilonnd{equation}
for some $C$ independent of $\beta$, $L$ and $n$. Here we set $w_{L,n}:=|u_{n}| u_{L,n}^{\beta-1}$.
Arguing as in the proof of Lemma $5.1$ in \cite{Apota} we can see that
\begin{equation}\label{UBu}
\|u_{n}\|_{L^{\infty}(\mathbb{R}^{N})}\leq K \mbox{ for all } n\in \mathbb{N}.
\varepsilonnd{equation}
Moreover, by interpolation, $(|u_{n}|)$ strongly converges in $L^{r}(\mathbb{R}^{N}, \mathbb{R})$ for all $r\in (2, \infty)$, and in view of the growth assumptions on $g$, also $g(\varepsilon_{n} x, |u_{n}|^{2})|u_{n}|$ strongly converges in the same Lebesgue spaces. \\
In what follows, we show that $|u_{n}|$ is a weak subsolution to
\begin{equation}\label{Kato0}
\left\{
\begin{array}{ll}
(-\Delta)^{s}v+V(\varepsilon_{n} x) v=\left(\frac{1}{|x|^{\mu}}*G(\varepsilon_{n} x, v^{2})\right)g(\varepsilon_{n} x, v^{2})v &\mbox{ in } \mathbb{R}^{N} \\
v\geq 0 \quad \mbox{ in } \mathbb{R}^{N}.
\varepsilonnd{array}
\right.
\varepsilonnd{equation}
Fix $\varphi\in C^{\infty}_{c}(\mathbb{R}^{N}, \mathbb{R})$ such that $\varphi\geq 0$, and we take $\psi_{\delta, n}=\frac{u_{n}}{u_{\delta, n}}\varphi$ as test function in \varepsilonqref{Pe}, where $u_{\delta,n}=\sqrt{|u_{n}|^{2}+\delta^{2}}$ for $\delta>0$. We note that $\psi_{\delta, n}\in H^{s}_{\varepsilon_{n}}$ for all $\delta>0$ and $n\in \mathbb{N}$.
Indeed, it is clear that
$$
\int_{\mathbb{R}^{N}} V(\varepsilon_{n} x) |\psi_{\delta,n}|^{2} dx\leq \int_{\supp(\varphi)} V(\varepsilon_{n} x)\varphi^{2} dx<\infty.
$$
Now, we show that $[\psi_{\delta, n}]_{A_{\varepsilon}}$ is finite. Let us observe that
\begin{align*}
\psi_{\delta,n}(x)-\psi_{\delta,n}(y)e^{\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)}&=\Bigl(\frac{u_{n}(x)}{u_{\delta,n}(x)}\Bigr)\varphi(x)-\Bigl(\frac{u_{n}(y)}{u_{\delta,n}(y)}\Bigr)\varphi(y)e^{\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)}\\
&=\Bigl[\Bigl(\frac{u_{n}(x)}{u_{\delta,n}(x)}\Bigr)-\Bigl(\frac{u_{n}(y)}{u_{\delta,n}(x)}\Bigr)e^{\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)}\Bigr]\varphi(x) \\
&+\Bigl[\varphi(x)-\varphi(y)\Bigr] \Bigl(\frac{u_{n}(y)}{u_{\delta,n}(x)}\Bigr) e^{\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)} \\
&+\Bigl(\frac{u_{n}(y)}{u_{\delta,n}(x)}-\frac{u_{n}(y)}{u_{\delta,n}(y)}\Bigr)\varphi(y) e^{\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)}.
\varepsilonnd{align*}
Then, by using $|z+w+k|^{2}\leq 4(|z|^{2}+|w|^{2}+|k|^{2})$ for all $z,w,k\in \mathbb{C}$, $|e^{\imath t}|=1$ for all $t\in \mathbb{R}$, $u_{\delta,n}\geq \delta$, $|\frac{u_{n}}{u_{\delta,n}}|\leq 1$, \varepsilonqref{UBu} and $|\sqrt{|z|^{2}+\delta^{2}}-\sqrt{|w|^{2}+\delta^{2}}|\leq ||z|-|w||$ for all $z, w\in \mathbb{C}$, we obtain that
\begin{align*}
&|\psi_{\delta,n}(x)-\psi_{\delta,n}(y)e^{\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)}|^{2} \\
&\leq \frac{4}{\delta^{2}}|u_{n}(x)-u_{n}(y)e^{\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)}|^{2}\|\varphi\|^{2}_{L^{\infty}(\mathbb{R}^{N})} +\frac{4}{\delta^{2}}|\varphi(x)-\varphi(y)|^{2} \|u_{n}\|^{2}_{L^{\infty}(\mathbb{R}^{N})} \\
&+\frac{4}{\delta^{4}} \|u_{n}\|^{2}_{L^{\infty}(\mathbb{R}^{N})} \|\varphi\|^{2}_{L^{\infty}(\mathbb{R}^{N})} |u_{\delta,n}(y)-u_{\delta,n}(x)|^{2} \\
&\leq \frac{4}{\delta^{2}}|u_{n}(x)-u_{n}(y)e^{\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)}|^{2}\|\varphi\|^{2}_{L^{\infty}(\mathbb{R}^{N})} +\frac{4K^{2}}{\delta^{2}}|\varphi(x)-\varphi(y)|^{2} \\
&+\frac{4K^{2}}{\delta^{4}} \|\varphi\|^{2}_{L^{\infty}(\mathbb{R}^{N})} ||u_{n}(y)|-|u_{n}(x)||^{2}.
\varepsilonnd{align*}
Since $u_{n}\in H^{s}_{\varepsilon_{n}}$, $|u_{n}|\in H^{s}(\mathbb{R}^{N}, \mathbb{R})$ (by Lemma \ref{DI}) and $\varphi\in C^{\infty}_{c}(\mathbb{R}^{N}, \mathbb{R})$, we conclude that $\psi_{\delta,n}\in H^{s}_{\varepsilon_{n}}$.
Then we get
\begin{align}\label{Kato1}
&\mathbb{R}e\Bigl[\iint_{\mathbb{R}^{2N}} \frac{(u_{n}(x)-u_{n}(y)e^{\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)})}{|x-y|^{N+2s}} \times \nonumber \\
&\quad \times \Bigl(\frac{\overline{u_{n}(x)}}{u_{\delta,n}(x)}\varphi(x)\!-\!\frac{\overline{u_{n}(y)}}{u_{\delta,n}(y)}\varphi(y)e^{-\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)} \Bigr) dx dy\Bigr]
+\int_{\mathbb{R}^{N}} V(\varepsilon_{n} x)\frac{|u_{n}|^{2}}{u_{\delta,n}}\varphi dx\nonumber\\
&=\int_{\mathbb{R}^{N}} \Bigl(\frac{1}{|x|^{\mu}}*G(\varepsilon_{n} x, |u_{n}|^{2})\Bigr) g(\varepsilon_{n} x, |u_{n}|^{2})\frac{|u_{n}|^{2}}{u_{\delta,n}}\varphi dx.
\varepsilonnd{align}
Now, we aim to pass to the limit as $\delta\rightarrow 0$ in \varepsilonqref{Kato1} to deduce that \varepsilonqref{Kato0} holds true.
Since $\mathbb{R}e(z)\leq |z|$ for all $z\in \mathbb{C}$ and $|e^{\imath t}|=1$ for all $t\in \mathbb{R}$, we have
\begin{align}\label{alves1}
&\mathbb{R}e\Bigl[(u_{n}(x)-u_{n}(y)e^{\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)}) \Bigl(\frac{\overline{u_{n}(x)}}{u_{\delta,n}(x)}\varphi(x)-\frac{\overline{u_{n}(y)}}{u_{\delta,n}(y)}\varphi(y)e^{-\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)} \Bigr)\Bigr] \nonumber\\
&=\mathbb{R}e\Bigl[\frac{|u_{n}(x)|^{2}}{u_{\delta,n}(x)}\varphi(x)+\frac{|u_{n}(y)|^{2}}{u_{\delta,n}(y)}\varphi(y)-\frac{u_{n}(x)\overline{u_{n}(y)}}{u_{\delta,n}(y)}\varphi(y)e^{-\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)} \nonumber \\
&-\frac{u_{n}(y)\overline{u_{n}(x)}}{u_{\delta,n}(x)}\varphi(x)e^{\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)}\Bigr] \nonumber \\
&\geq \Bigl[\frac{|u_{n}(x)|^{2}}{u_{\delta,n}(x)}\varphi(x)+\frac{|u_{n}(y)|^{2}}{u_{\delta,n}(y)}\varphi(y)-|u_{n}(x)|\frac{|u_{n}(y)|}{u_{\delta,n}(y)}\varphi(y)-|u_{n}(y)|\frac{|u_{n}(x)|}{u_{\delta,n}(x)}\varphi(x) \Bigr].
\varepsilonnd{align}
Let us note that
\begin{align}\label{alves2}
&\frac{|u_{n}(x)|^{2}}{u_{\delta,n}(x)}\varphi(x)+\frac{|u_{n}(y)|^{2}}{u_{\delta,n}(y)}\varphi(y)-|u_{n}(x)|\frac{|u_{n}(y)|}{u_{\delta,n}(y)}\varphi(y)-|u_{n}(y)|\frac{|u_{n}(x)|}{u_{\delta,n}(x)}\varphi(x) \nonumber\\
&= \frac{|u_{n}(x)|}{u_{\delta,n}(x)}(|u_{n}(x)|-|u_{n}(y)|)\varphi(x)-\frac{|u_{n}(y)|}{u_{\delta,n}(y)}(|u_{n}(x)|-|u_{n}(y)|)\varphi(y) \nonumber\\
&=\Bigl[\frac{|u_{n}(x)|}{u_{\delta,n}(x)}(|u_{n}(x)|-|u_{n}(y)|)\varphi(x)-\frac{|u_{n}(x)|}{u_{\delta,n}(x)}(|u_{n}(x)|-|u_{n}(y)|)\varphi(y)\Bigr] \nonumber\\
&\quad +\Bigl(\frac{|u_{n}(x)|}{u_{\delta,n}(x)}-\frac{|u_{n}(y)|}{u_{\delta,n}(y)} \Bigr) (|u_{n}(x)|-|u_{n}(y)|)\varphi(y) \nonumber\\
&=\frac{|u_{n}(x)|}{u_{\delta,n}(x)}(|u_{n}(x)|-|u_{n}(y)|)(\varphi(x)-\varphi(y)) \nonumber \\
&\quad +\Bigl(\frac{|u_{n}(x)|}{u_{\delta,n}(x)}-\frac{|u_{n}(y)|}{u_{\delta,n}(y)} \Bigr) (|u_{n}(x)|-|u_{n}(y)|)\varphi(y) \nonumber\\
&\geq \frac{|u_{n}(x)|}{u_{\delta,n}(x)}(|u_{n}(x)|-|u_{n}(y)|)(\varphi(x)-\varphi(y))
\varepsilonnd{align}
where in the last inequality we used the fact that
$$
\left(\frac{|u_{n}(x)|}{u_{\delta,n}(x)}-\frac{|u_{n}(y)|}{u_{\delta,n}(y)} \right) (|u_{n}(x)|-|u_{n}(y)|)\varphi(y)\geq 0
$$
because
$$
h(t)=\frac{t}{\sqrt{t^{2}+\delta^{2}}} \mbox{ is increasing for } t\geq 0 \quad \mbox{ and } \quad \varphi\geq 0 \mbox{ in }\mathbb{R}^{N}.
$$
Since
\begin{align*}
&\frac{|\frac{|u_{n}(x)|}{u_{\delta,n}(x)}(|u_{n}(x)|-|u_{n}(y)|)(\varphi(x)-\varphi(y))|}{|x-y|^{N+2s}}\\
&\quad \leq \frac{||u_{n}(x)|-|u_{n}(y)||}{|x-y|^{\frac{N+2s}{2}}} \frac{|\varphi(x)-\varphi(y)|}{|x-y|^{\frac{N+2s}{2}}}\in L^{1}(\mathbb{R}^{2N}),
\varepsilonnd{align*}
and $\frac{|u_{n}(x)|}{u_{\delta,n}(x)}\rightarrow 1$ a.e. in $\mathbb{R}^{N}$ as $\delta\rightarrow 0$,
we can use \varepsilonqref{alves1}, \varepsilonqref{alves2} and the Dominated Convergence Theorem to deduce that
\begin{align}\label{Kato2}
&\limsup_{\delta\rightarrow 0} \mathbb{R}e\Bigl[\iint_{\mathbb{R}^{2N}} \frac{(u_{n}(x)-u_{n}(y)e^{\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)})}{|x-y|^{N+2s}} \times \nonumber \\
&\quad \times \Bigl(\frac{\overline{u_{n}(x)}}{u_{\delta,n}(x)}\varphi(x)-\frac{\overline{u_{n}(y)}}{u_{\delta,n}(y)}\varphi(y)e^{-\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)} \Bigr) dx dy\Bigr] \nonumber\\
&\geq \limsup_{\delta\rightarrow 0} \iint_{\mathbb{R}^{2N}} \frac{|u_{n}(x)|}{u_{\delta,n}(x)}(|u_{n}(x)|-|u_{n}(y)|)(\varphi(x)-\varphi(y)) \frac{dx dy}{|x-y|^{N+2s}} \nonumber\\
&=\iint_{\mathbb{R}^{2N}} \frac{(|u_{n}(x)|-|u_{n}(y)|)(\varphi(x)-\varphi(y))}{|x-y|^{N+2s}} dx dy.
\varepsilonnd{align}
On the other hand, from the Dominated Convergence Theorem (we note that $\frac{|u_{n}|^{2}}{u_{\delta, n}}\leq |u_{n}|$, $\varphi\in C^{\infty}_{c}(\mathbb{R}^{N}, \mathbb{R})$ and $\tilde{K}_{\varepsilon}(u_{n})$ is bounded in view of Lemma \ref{lemK}) we can infer that
\begin{equation}\label{Kato3}
\lim_{\delta\rightarrow 0} \int_{\mathbb{R}^{N}} V(\varepsilon_{n} x)\frac{|u_{n}|^{2}}{u_{\delta,n}}\varphi dx=\int_{\mathbb{R}^{N}} V(\varepsilon_{n} x)|u_{n}|\varphi dx
\varepsilonnd{equation}
and
\begin{align}\label{Kato4}
&\lim_{\delta\rightarrow 0} \int_{\mathbb{R}^{N}} \left(\frac{1}{|x|^{\mu}}*G(\varepsilon_{n} x, |u_{n}|^{2})\right) g(\varepsilon_{n} x, |u_{n}|^{2})\frac{|u_{n}|^{2}}{u_{\delta,n}}\varphi dx\nonumber \\
&=\int_{\mathbb{R}^{N}} \left(\frac{1}{|x|^{\mu}}*G(\varepsilon_{n} x, |u_{n}|^{2})\right) g(\varepsilon_{n} x, |u_{n}|^{2}) |u_{n}|\varphi dx.
\varepsilonnd{align}
Taking into account \varepsilonqref{Kato1}, \varepsilonqref{Kato2}, \varepsilonqref{Kato3} and \varepsilonqref{Kato4} we can see that
\begin{align*}
&\iint_{\mathbb{R}^{2N}} \frac{(|u_{n}(x)|-|u_{n}(y)|)(\varphi(x)-\varphi(y))}{|x-y|^{N+2s}} dx dy+\int_{\mathbb{R}^{N}} V(\varepsilon_{n} x)|u_{n}|\varphi dx\\
&\leq \int_{\mathbb{R}^{N}} \left(\frac{1}{|x|^{\mu}}*G(\varepsilon_{n} x, |u_{n}|^{2})\right) g(\varepsilon_{n} x, |u_{n}|^{2}) |u_{n}|\varphi dx
\varepsilonnd{align*}
for any $\varphi\in C^{\infty}_{c}(\mathbb{R}^{N}, \mathbb{R})$ such that $\varphi\geq 0$. Then $|u_{n}|$ is a weak subsolution to \varepsilonqref{Kato0}.
By using $(V_{1})$, $u_{n}\in \B$ for all $n$ big enough, and Lemma \ref{lemK}, it is clear that $v_{n}=|u_{n}|(\cdot+\tilde{y}_{n})$ solves
\begin{equation}\label{Pkat}
(-\Delta)^{s} v_{n} + V_{0}v_{n}\leq C_{0} g(\varepsilon_{n} x+\varepsilon_{n}\tilde{y}_{n}, v_{n}^{2})v_{n} \mbox{ in } \mathbb{R}^{N}.
\varepsilonnd{equation}
Let us denote by $z_{n}\in H^{s}(\mathbb{R}^{N}, \mathbb{R})$ the unique solution to
\begin{equation}\label{US}
(-\Delta)^{s} z_{n} + V_{0}z_{n}= C_{0}g_{n} \mbox{ in } \mathbb{R}^{N},
\varepsilonnd{equation}
where
$$
g_{n}:=g(\varepsilon_{n} x+\varepsilon_{n}\tilde{y}_{n}, v_{n}^{2})v_{n}\in L^{r}(\mathbb{R}^{N}, \mathbb{R}) \quad \forall r\in [2, \infty].
$$
Since \varepsilonqref{UBu} yields $\|v_{n}\|_{L^{\infty}(\mathbb{R}^{N})}\leq C$ for all $n\in \mathbb{N}$, by interpolation we know that $v_{n}\rightarrow v$ strongly converges in $L^{r}(\mathbb{R}^{N}, \mathbb{R})$ for all $r\in (2, \infty)$, for some $v\in L^{r}(\mathbb{R}^{N}, \mathbb{R})$, and by the growth assumptions on $f$, we can see that also $g_{n}\rightarrow f(v^{2})v$ in $L^{r}(\mathbb{R}^{N}, \mathbb{R})$ and $\|g_{n}\|_{L^{\infty}(\mathbb{R}^{N})}\leq C$ for all $n\in \mathbb{N}$.
Since $z_{n}=\mathcal{K}*(C_{0}g_{n})$, where $\mathcal{K}$ is the Bessel kernel (see \cite{FQT}), we can argue as in \cite{AM} to infer that $|z_{n}(x)|\rightarrow 0$ as $|x|\rightarrow \infty$ uniformly with respect to $n\in \mathbb{N}$.
On the other hand, $v_{n}$ satisfies \varepsilonqref{Pkat} and $z_{n}$ solves \varepsilonqref{US} so a simple comparison argument shows that $0\leq v_{n}\leq z_{n}$ a.e. in $\mathbb{R}^{N}$ and for all $n\in \mathbb{N}$. This means that $v_{n}(x)\rightarrow 0$ as $|x|\rightarrow \infty$ uniformly in $n\in \mathbb{N}$.
\varepsilonnd{proof}
| 7,635 | 38,947 |
en
|
train
|
0.43.10
|
\noindent
At this point we have all ingredients to give the proof of Theorem \ref{thm1}.
\begin{proof}
By using Lemma \ref{prop3.3} we can find a sequence $(\tilde{y}_{n})\subset \mathbb{R}^{N}$ such that $\varepsilon_{n}\tilde{y}_{n}\rightarrow y_{0}$ for some $y_{0} \in \Lambda$ such that $V(y_{0})=V_{0}$.
Then there exists $r>0$ such that, for some subsequence still denoted by itself, we have $B_{r}(\tilde{y}_{n})\subset \Lambda$ for all $n\in \mathbb{N}$.
Hence $B_{\frac{r}{\varepsilon_{n}}}(\tilde{y}_{n})\subset \Lambda_{\varepsilon_{n}}$ for all $n\in \mathbb{N}$, which gives
$$
\mathbb{R}^{N}\setminus \Lambda_{\varepsilon_{n}}\subset \mathbb{R}^{N} \setminus B_{\frac{r}{\varepsilon_{n}}}(\tilde{y}_{n}) \mbox{ for any } n\in \mathbb{N}.
$$
In view of Lemma \ref{moser}, we know that there exists $R>0$ such that
$$
v_{n}(x)<a \mbox{ for } |x|\geq R \mbox{ and } n\in \mathbb{N},
$$
where $v_{n}(x)=|u_{\varepsilon_{n}}|(x+ \tilde{y}_{n})$.
Then $|u_{\varepsilon_{n}}(x)|<a$ for any $x\in \mathbb{R}^{N}\setminus B_{R}(\tilde{y}_{n})$ and $n\in \mathbb{N}$. Moreover, there exists $\nu \in \mathbb{N}$ such that for any $n\geq \nu$ and $r/\varepsilon_{n}>R$ it holds
$$
\mathbb{R}^{N}\setminus \Lambda_{\varepsilon_{n}}\subset \mathbb{R}^{N} \setminus B_{\frac{r}{\varepsilon_{n}}}(\tilde{y}_{n})\subset \mathbb{R}^{N}\setminus B_{R}(\tilde{y}_{n}),
$$
which gives $|u_{\varepsilon_{n}}(x)|<a$ for any $x\in \mathbb{R}^{N}\setminus \Lambda_{\varepsilon_{n}}$ and $n\geq \nu$. \\
Therefore, there exists $\varepsilon_{0}>0$ such that problem \varepsilonqref{R} admits a nontrivial solution $u_{\varepsilon}$ for all $\varepsilon\in (0, \varepsilon_{0})$. Then $H^{s}_{\e}at{u}_{\varepsilon}(x)=u_{\varepsilon}(x/\varepsilon)$ is a solution to (\ref{P}).
Finally, we study the behavior of the maximum points of $|u_{\varepsilon_{n}}|$. In view of $(g_1)$, there exists $\gamma\in (0,a)$ such that
\begin{align}\label{4.18HZ}
g(\varepsilon x, t^{2})t^{2}\leq \frac{V_{0}}{\varepsilonll_{0}}t^{2}, \mbox{ for all } x\in \mathbb{R}^{N}, |t|\leq \gamma.
\varepsilonnd{align}
Using a similar discussion above, we can take $R>0$ such that
\begin{align}\label{4.19HZ}
\|u_{\varepsilon_{n}}\|_{L^{\infty}(B^{c}_{R}(\tilde{y}_{n}))}<\gamma.
\varepsilonnd{align}
Up to a subsequence, we may also assume that
\begin{align}\label{4.20HZ}
\|u_{\varepsilon_{n}}\|_{L^{\infty}(B_{R}(\tilde{y}_{n}))}\geq \gamma.
\varepsilonnd{align}
Otherwise, if \varepsilonqref{4.20HZ} does not hold, then $\|u_{\varepsilon_{n}}\|_{L^{\infty}(\mathbb{R}^{N})}< \gamma$ and by using $J_{\varepsilon_{n}}'(u_{\varepsilon_{n}})=0$, \varepsilonqref{4.18HZ}, Lemma \ref{DI} and
$$
\left\|\frac{1}{|x|^{\mu}}*G(\varepsilon x, |u_{n}|^{2})\right\|_{L^{\infty}(\mathbb{R}^{N})}<C_{0},
$$
we have
\begin{align*}
[|u_{\varepsilon_{n}}|]^{2}+\int_{\mathbb{R}^{N}}V_{0}|u_{\varepsilon_{n}}|^{2}dx\leq \|u_{\varepsilon_{n}}\|^{2}_{\varepsilon_{n}}&=\int_{\mathbb{R}^{N}} \tilde{K}_{\varepsilon}(u_{\varepsilon_{n}}) g_{\varepsilon_{n}}(x, |u_{\varepsilon_{n}}|^{2})|u_{\varepsilon_{n}}|^{2}\,dx\\
&\leq \frac{C_{0}V_{0}}{\varepsilonll_{0}}\int_{\mathbb{R}^{N}}|u_{\varepsilon_{n}}|^{2}\, dx
\varepsilonnd{align*}
and being $\frac{C_{0}}{\varepsilonll_{0}}<\frac{1}{2}$ we deduce that $\||u_{\varepsilon_{n}}|\|_{H^{s}(\mathbb{R}^{N})}=0$ which is impossible.
From \varepsilonqref{4.19HZ} and \varepsilonqref{4.20HZ}, it follows that the maximum points $p_{n}$ of $|u_{\varepsilon_{n}}|$ belong to $B_{R}(\tilde{y}_{n})$, that is $p_{n}=\tilde{y}_{n}+q_{n}$ for some $q_{n}\in B_{R}$. Since $H^{s}_{\e}at{u}_{n}(x)=u_{\varepsilon_{n}}(x/\varepsilon_{n})$ is a solution to \varepsilonqref{P}, we can see that the maximum point $\varepsilonta_{\varepsilon_{n}}$ of $|H^{s}_{\e}at{u}_{n}|$ is given by $\varepsilonta_{\varepsilon_{n}}=\varepsilon_{n}\tilde{y}_{n}+\varepsilon_{n}q_{n}$. Taking into account $q_{n}\in B_{R}$, $\varepsilon_{n}\tilde{y}_{n}\rightarrow y_{0}$ and $V(y_{0})=V_{0}$ and the continuity of $V$, we can infer that
$$
\lim_{n\rightarrow \infty} V(\varepsilonta_{\varepsilon_{n}})=V_{0}.
$$
Finally, we give a decay estimate for $|H^{s}_{\e}at{u}_{n}|$. We follow some arguments used in \cite{A6}.\\
Invoking Lemma $4.3$ in \cite{FQT}, we can find a function $w$ such that
\begin{align}\label{HZ1}
0<w(x)\leq \frac{C}{1+|x|^{N+2s}},
\varepsilonnd{align}
and
\begin{align}\label{HZ2}
(-\Delta)^{s} w+\frac{V_{0}}{2}w\geq 0 \mbox{ in } \mathbb{R}^{N}\setminus B_{R_{1}}
\varepsilonnd{align}
for some suitable $R_{1}>0$. Using Lemma \ref{moser}, we know that $v_{n}(x)\rightarrow 0$ as $|x|\rightarrow \infty$ uniformly in $n\in \mathbb{N}$, so there exists $R_{2}>0$ such that
\begin{equation}\label{hzero}
h_{n}=C_{0}g(\varepsilon_{n}x+\varepsilon_{n}\tilde{y}_{n}, v_{n}^{2})v_{n}\leq \frac{C_{0}V_{0}}{\varepsilonll_{0}}v_{n}\leq \frac{V_{0}}{2}v_{n} \mbox{ in } B_{R_{2}}^{c}.
\varepsilonnd{equation}
Let us denote by $w_{n}$ the unique solution to
$$
(-\Delta)^{s}w_{n}+V_{0}w_{n}=h_{n} \mbox{ in } \mathbb{R}^{N}.
$$
Then $w_{n}(x)\rightarrow 0$ as $|x|\rightarrow \infty$ uniformly in $n\in \mathbb{N}$, and by comparison $0\leq v_{n}\leq w_{n}$ in $\mathbb{R}^{N}$. Moreover, in view of \varepsilonqref{hzero}, it holds
\begin{align*}
(-\Delta)^{s}w_{n}+\frac{V_{0}}{2}w_{n}=h_{n}-\frac{V_{0}}{2}w_{n}\leq 0 \mbox{ in } B_{R_{2}}^{c}.
\varepsilonnd{align*}
Take $R_{3}=\max\{R_{1}, R_{2}\}$ and we define
\begin{align}\label{HZ4}
a=\inf_{B_{R_{3}}} w>0 \mbox{ and } \tilde{w}_{n}=(b+1)w-a w_{n}.
\varepsilonnd{align}
where $b=\sup_{n\in \mathbb{N}} \|w_{n}\|_{L^{\infty}(\mathbb{R}^{N})}<\infty$.
We aim to prove that
\begin{equation}\label{HZ5}
\tilde{w}_{n}\geq 0 \mbox{ in } \mathbb{R}^{N}.
\varepsilonnd{equation}
Let us note that
\begin{align}
&\lim_{|x|\rightarrow \infty} \tilde{w}_{n}(x)=0 \mbox{ uniformly in } n\in \mathbb{N}, \label{HZ0N} \\
&\tilde{w}_{n}\geq ba+w-ba>0 \mbox{ in } B_{R_{3}} \label{HZ0},\\
&(-\Delta)^{s} \tilde{w}_{n}+\frac{V_{0}}{2}\tilde{w}_{n}\geq 0 \mbox{ in } \mathbb{R}^{N}\setminus B_{R_{3}} \label{HZ00}.
\varepsilonnd{align}
We argue by contradiction, and we assume that there exists a sequence $(\bar{x}_{j, n})\subset \mathbb{R}^{N}$ such that
\begin{align}\label{HZ6}
\inf_{x\in \mathbb{R}^{N}} \tilde{w}_{n}(x)=\lim_{j\rightarrow \infty} \tilde{w}_{n}(\bar{x}_{j, n})<0.
\varepsilonnd{align}
From (\ref{HZ0N}) it follows that $(\bar{x}_{j, n})$ is bounded, and, up to subsequence, we may assume that there exists $\bar{x}_{n}\in \mathbb{R}^{N}$ such that $\bar{x}_{j, n}\rightarrow \bar{x}_{n}$ as $j\rightarrow \infty$.
In view of (\ref{HZ6}) we can see that
\begin{align}\label{HZ7}
\inf_{x\in \mathbb{R}^{N}} \tilde{w}_{n}(x)= \tilde{w}_{n}(\bar{x}_{n})<0.
\varepsilonnd{align}
By using the minimality of $\bar{x}_{n}$ and the integral representation formula for the fractional Laplacian \cite{DPV}, we can see that
\begin{align}\label{HZ8}
(-\Delta)^{s}\tilde{w}_{n}(\bar{x}_{n})=\frac{C(N, s)}{2} \int_{\mathbb{R}^{N}} \frac{2\tilde{w}_{n}(\bar{x}_{n})-\tilde{w}_{n}(\bar{x}_{n}+\xi)-\tilde{w}_{n}(\bar{x}_{n}-\xi)}{|\xi|^{N+2s}} d\xi\leq 0.
\varepsilonnd{align}
Putting together (\ref{HZ0}) and (\ref{HZ6}), we have $\bar{x}_{n}\in \mathbb{R}^{N}\setminus B_{R_{3}}$.
This fact combined with (\ref{HZ7}) and (\ref{HZ8}) yields
$$
(-\Delta)^{s} \tilde{w}_{n}(\bar{x}_{n})+\frac{V_{0}}{2}\tilde{w}_{n}(\bar{x}_{n})<0,
$$
which gives a contradiction in view of (\ref{HZ00}).
As a consequence (\ref{HZ5}) holds true, and by using (\ref{HZ1}) and $v_{n}\leq w_{n}$ we can deduce that
\begin{align*}
0\leq v_{n}(x)\leq w_{n}(x)\leq \frac{(b+1)}{a}w(x)\leq \frac{\tilde{C}}{1+|x|^{N+2s}} \mbox{ for all } n\in \mathbb{N}, x\in \mathbb{R}^{N},
\varepsilonnd{align*}
for some constant $\tilde{C}>0$. Recalling the definition of $v_{n}$, we can obtain that
\begin{align*}
|H^{s}_{\e}at{u}_{n}(x)|&=\left|u_{\varepsilon_{n}}\left(\frac{x}{\varepsilon_{n}}\right)\right|=v_{n}\left(\frac{x}{\varepsilon_{n}}-\tilde{y}_{n}\right) \\
&\leq \frac{\tilde{C}}{1+|\frac{x}{\varepsilon_{n}}-\tilde{y}_{\varepsilon_{n}}|^{N+2s}} \\
&=\frac{\tilde{C} \varepsilon_{n}^{N+2s}}{\varepsilon_{n}^{N+2s}+|x- \varepsilon_{n} \tilde{y}_{\varepsilon_{n}}|^{N+2s}} \\
&\leq \frac{\tilde{C} \varepsilon_{n}^{N+2s}}{\varepsilon_{n}^{N+2s}+|x-\varepsilonta_{\varepsilon_{n}}|^{N+2s}}.
\varepsilonnd{align*}
\varepsilonnd{proof}
\noindent {\bf Acknowledgements.}
The author thanks Claudianor O. Alves and Hoai-Minh Nguyen for delightful and pleasant discussions about the results of this work.
\begin{thebibliography}{777}
\bibitem{Ack}
N. Ackermann,
On a periodic Schr\"odinger equation with nonlocal superlinear part,
{\it Math. Z.} {\bf 248, no. 2} (2004), 423-443.
\bibitem{AF}
C.O. Alves and G.M. Figueiredo,
Multiple Solutions for a Semilinear Elliptic Equation with Critical Growth and Magnetic Field,
{\it Milan J. Math.} {\bf 82, no. 2} (2014), 389-405.
\bibitem{AFF}
C.O. Alves, G.M. Figueiredo and M.F. Furtado,
Multiple solutions for a nonlinear Schr\"odinger equation with magnetic fields,
{\it Comm. Partial Differential Equations} {\bf 36, no. 9} (2011), 1565-1586.
\bibitem{AFY}
C.O. Alves, G.M. Figueiredo and M.Yang,
Multiple semiclassical solutions for a nonlinear Choquard equation with magnetic field,
{\it Asymptot. Anal.} {\bf 96, no. 2} (2016), 135-159.
\bibitem{AM}
C. O. Alves and O. H. Miyagaki,
Existence and concentration of solution for a class of fractional elliptic equation in $\mathbb{R}^{N}$ via penalization method,
{\it Calc. Var. Partial Differential Equations } {\bf 55, no. 3} (2016), Art. 47, 19 pp.
\bibitem{AR}
A. Ambrosetti P. H. and Rabinowitz,
Dual variational methods in critical point theory and applications,
{\it J. Funct. Anal.} {\bf 14} (1973), 349-381.
\bibitem{A6}
V. Ambrosio,
Concentrating solutions for a class of nonlinear fractional Schr\"odinger equations in $\mathbb{R}^{N}$,
{\it Rev. Mat. Iberoam.} {\it (in press)} (2017), arXiv:1612.02388.
\bibitem{Apota}
V. Ambrosio,
Multiplicity and concentration results for a fractional Choquard equation via penalization method,
{\it Potential Analysis} {\it (in press)}, DOI: 10.1007/s11118-017-9673-3.
| 4,037 | 38,947 |
en
|
train
|
0.43.11
|
\varepsilonnd{proof}
\noindent {\bf Acknowledgements.}
The author thanks Claudianor O. Alves and Hoai-Minh Nguyen for delightful and pleasant discussions about the results of this work.
\begin{thebibliography}{777}
\bibitem{Ack}
N. Ackermann,
On a periodic Schr\"odinger equation with nonlocal superlinear part,
{\it Math. Z.} {\bf 248, no. 2} (2004), 423-443.
\bibitem{AF}
C.O. Alves and G.M. Figueiredo,
Multiple Solutions for a Semilinear Elliptic Equation with Critical Growth and Magnetic Field,
{\it Milan J. Math.} {\bf 82, no. 2} (2014), 389-405.
\bibitem{AFF}
C.O. Alves, G.M. Figueiredo and M.F. Furtado,
Multiple solutions for a nonlinear Schr\"odinger equation with magnetic fields,
{\it Comm. Partial Differential Equations} {\bf 36, no. 9} (2011), 1565-1586.
\bibitem{AFY}
C.O. Alves, G.M. Figueiredo and M.Yang,
Multiple semiclassical solutions for a nonlinear Choquard equation with magnetic field,
{\it Asymptot. Anal.} {\bf 96, no. 2} (2016), 135-159.
\bibitem{AM}
C. O. Alves and O. H. Miyagaki,
Existence and concentration of solution for a class of fractional elliptic equation in $\mathbb{R}^{N}$ via penalization method,
{\it Calc. Var. Partial Differential Equations } {\bf 55, no. 3} (2016), Art. 47, 19 pp.
\bibitem{AR}
A. Ambrosetti P. H. and Rabinowitz,
Dual variational methods in critical point theory and applications,
{\it J. Funct. Anal.} {\bf 14} (1973), 349-381.
\bibitem{A6}
V. Ambrosio,
Concentrating solutions for a class of nonlinear fractional Schr\"odinger equations in $\mathbb{R}^{N}$,
{\it Rev. Mat. Iberoam.} {\it (in press)} (2017), arXiv:1612.02388.
\bibitem{Apota}
V. Ambrosio,
Multiplicity and concentration results for a fractional Choquard equation via penalization method,
{\it Potential Analysis} {\it (in press)}, DOI: 10.1007/s11118-017-9673-3.
\bibitem{AD}
V. Ambrosio and P. d'Avenia,
Nonlinear fractional magnetic Schr\"odinger equation: existence and multiplicity,
{\it J. Differential Equations} {\bf 264, no. 5} (2018), 3336-3368.
\bibitem{AS}
G. Arioli and A. Szulkin,
A semilinear Schr\"odinger equation in the presence of a magnetic field,
{\it Arch. Ration. Mech. Anal.} {\bf 170, no. 4} (2003), 277-295.
\bibitem{Cingolani}
S. Cingolani,
Semiclassical stationary states of nonlinear Schr\"odinger equations with an external magnetic field,
{\it J. Differential Equations} {\bf 188, no. 1} (2003), 52-79.
\bibitem{CS}
S. Cingolani, S. Secchi,
Semiclassical states for NLS equations with magnetic potentials having polynomial growths,
{\it J. Math. Phys.} {\bf 46, no. 5} (2005), 19 pp.
\bibitem{CSS}
S. Cingolani, S. Secchi and M. Squassina,
Semi-classical limit for Schro\"dinger equations with magnetic field and Hartree-type nonlinearities,
{\it Proc. Roy. Soc. Edinburgh Sect. A} {\bf 140, no. 5} (2010), 973-1009.
\bibitem{DSS}
P. d'Avenia, G. Siciliano, and M. Squassina,
On fractional Choquard equations,
{\it Math. Models Methods Appl. Sci.} {\bf 25, no. 8} (2015), 1447-1476.
\bibitem{DS}
P. d'Avenia and M. Squassina,
Ground states for fractional magnetic operators,
{\it ESAIM Control Optim. Calc. Var.} {\bf 24, no. 1} (2018), 1-24.
\bibitem{DPF}
M. Del Pino and P.L. Felmer,
Local Mountain Pass for semilinear elliptic problems in unbounded domains,
{\it Calc. Var. Partial Differential Equations} {\bf 4, no. 2} (1996), 121-137.
\bibitem{DPMV}
S. Dipierro, M. Medina and E. Valdinoci,
Fractional elliptic problems with critical growth in the whole of $\mathbb{R}^{n}$,
Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], 15. Edizioni della Normale, Pisa, 2017. viii+152 pp.
\bibitem{DPV}
E. Di Nezza, G. Palatucci and E. Valdinoci,
Hitchhiker's guide to the fractional Sobolev spaces,
{\it Bull. Sci. math.} {\bf 136, no. 5} (2012), 521-573.
\bibitem{EL}
M. Esteban and P.L. Lions,
Stationary solutions of nonlinear Schr\"odinger equations with an external magnetic field,
Partial differential equations and the calculus of variations, Vol. I of {\it Progr. Nonlinear Differential Equations Appl.} Birkh\"auser Boston, Boston, MA (1989) 401-449.
\bibitem{FQT}
P. Felmer, A. Quaas and J. Tan,
Positive solutions of the nonlinear Schr\"odinger equation with the fractional Laplacian,
{\it Proc. Roy. Soc. Edinburgh Sect. A} {\bf 142, no. 6} (2012), 1237-1262.
\bibitem{FPV}
A. Fiscella, A. Pinamonti and E. Vecchi,
Multiplicity results for magnetic fractional problems,
{\it J. Differential Equations} {\bf 263, no. 8} (2017), 4617-4633.
\bibitem{FL}
R.L. Frank and E. Lenzmann,
On ground states for the $L^2$-critical boson star equation, {\it submitted} (2009), arXiv:0910.2721.
\bibitem{I}
T. Ichinose,
Magnetic relativistic Schr\"odinger operators and imaginary-time path integrals,
Mathematical physics, spectral theory and stochastic analysis, {\it Oper. Theory Adv. Appl.} 232, Birkh\"auser/Springer, Basel, (2013), 247-297.
\bibitem{Kato}
T. Kato,
Schr\"odinger operators with singular potentials,
Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces (Jerusalem, 1972), {\it Israel J. Math.} {\bf 13} (1973), 135-148.
\bibitem{K}
K. Kurata,
Existence and semi-classical limit of the least energy solution to a nonlinear Schr\"odinger equation with electromagnetic fields,
{\it Nonlinear Anal.} {\bf 41, no. 5-6} (2000), 763-778.
\bibitem{LaL}
L.D. Landau and E.M. Lifshitz,
Quantum mechanics, Pergamon Press, (1977).
\bibitem{Lieb}
E. H. Lieb
Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation,
{\it Stud. Appl. Math.} {\bf 57, no. 2} (1976/77), 93-105.
\bibitem{LL}
E. H. Lieb and M. Loss,
Analysis,
Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, 1997. xviii+278 pp.
\bibitem{LS}
E. H. Lieb and B. Simon,
The Hartree-Fock theory for Coulomb systems,
{\it Comm. Math. Phys.} {\bf 53, no. 3} (1977), 185-194.
\bibitem{Lions}
P.L. Lions,
The Choquard equation and related questions,
{\it Nonlinear Anal.} {\bf 4, no. 6} (1980), 1063-1072.
\bibitem{MZ}
L. Ma and L.Zhao
Classification of positive solitary solutions of the nonlinear Choquard equation,
{\it Arch. Ration. Mech. Anal.} {\bf 195, no. 2} (2010), 455-467.
\bibitem{MPSZ}
X. Mingqi, P. Pucci, M. Squassina and B. Zhang,
Nonlocal Schr\"odinger-Kirchhoff equations with external magnetic field,
{\it Discrete Contin. Dyn. Syst.} {\bf 37, no. 3} (2017), 503-521.
\bibitem{MBRS}
G. Molica Bisci, V. R\u{a}dulescu and R. Servadei,
Variational Methods for Nonlocal Fractional Problems,
Cambridge University Press, 162 Cambridge, 2016.
\bibitem{Moser}
J. Moser,
A new proof of De Giorgi's theorem concerning the regularity problem for elliptic differential equations,
{\it Comm. Pure Appl. Math.} {\bf 13} (1960), 457-468.
\bibitem{MVS1}
V. Moroz and J. Van Schaftingen,
Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics,
{\it J. Funct. Anal.} {\bf 265, no. 2} (2013), 153-84.
\bibitem{MVS2}
V. Moroz and J. Van Schaftingen,
Existence of groundstates for a class of nonlinear Choquard equations,
{\it Trans. Am. Math. Soc.} {\bf 367, no. 9} (2015), 6557-6579.
\bibitem{MVS3}
V. Moroz and J. Van Schaftingen,
A guide to the Choquard equation,
{\it J. Fixed Point Theory Appl.} {\bf 19, no. 1} (2017), 773-813.
\bibitem{NPSV}
H. M. Nguyen, A. Pinamonti, M. Squassina and E. Vecchi,
New characterization of magnetic Sobolev spaces,
{\it Adv. Nonlinear Anal.} {\bf 7, no. 2} (2018), 227-245.
\bibitem{PP}
G. Palatucci and A. Pisante,
Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces,
{\it Calc. Var. Partial Differential Equations} {\bf 50, no. 3-4} (2014), 799-829.
\bibitem{Pek}
S. Pekar,
Untersuchung \"uber die Elektronentheorie der Kristalle (Akademie Verlag, Berlin 1954).
\bibitem{Pen}
R. Penrose,
Quantum computation, entanglement and state reduction,
{\it R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci.} {\bf 356, no. 1743} (1998), 1927-1939.
\bibitem{PSV1}
A. Pinamonti, M. Squassina and E. Vecchi,
The Maz'ya-Shaposhnikova limit in the magnetic setting,
{\it J. Math. Anal. Appl.} {\bf 449, no. 2} (2017), 1152-1159.
\bibitem{PSV2}
A. Pinamonti, M. Squassina and E. Vecchi,
Magnetic BV functions and the Bourgain-Brezis-Mironescu formula,
to appear in {\it Adv. Calc. Var.}, DOI: 10.1515/acv-2017-0019.
\bibitem{SGY}
Z. Shen, F. Gao, and M. Yang,
Ground states for nonlinear fractional Choquard equations with general nonlinearities,
{\it Math. Methods Appl. Sci.} {\bf 39, no. 14} (2016), 4082-4098.
\bibitem{SV}
M. Squassina, B. Volzone,
Bourgain-Brezis-Mironescu formula for magnetic operators,
{\it C. R. Math. Acad. Sci. Paris} {\bf 354, no. 8} (2016), 825-831.
\bibitem{W}
M. Willem,
Minimax theorems,
Progress in Nonlinear Differential Equations and their Applications, 24. Birkh\"auser Boston, Inc., Boston, MA, 1996. x+162 pp.
\bibitem{ZSZ}
B. Zhang, M. Squassina and X. Zhang,
Fractional NLS equations with magnetic field, critical frequency and critical growth,
{\it Manuscripta Math.} {\bf 155, no. 1-2} (2018), 115-140.
\varepsilonnd{thebibliography}
\varepsilonnd{document}
| 3,954 | 38,947 |
en
|
train
|
0.44.0
|
\begin{document}
\title{Quantum Dissipation and Decoherence via Interaction with
Low-Dimensional Chaos: a Feynman-Vernon Approach}
\author{M.V.S. Bonan\c{c}a and M.A.M. de Aguiar}
\affiliation{Instituto de F\'isica 'Gleb Wataghin',
Universidade Estadual de Campinas, \\
Caixa Postal 6165, 13083-970 Campinas, S\~ao Paulo, Brazil}
\begin{abstract}
We study the effects of dissipation and decoherence induced on a
harmonic oscillator by the coupling to a chaotic system with two
degrees of freedom. Using the Feynman-Vernon approach and treating
the chaotic system semiclassically we show that the effects of the
low dimensional chaotic environment are in many ways similar to
those produced by thermal baths. The classical correlation and
response functions play important roles in both classical and
quantum formulations. Our results are qualitatively similar to the
high temperature regime of the Caldeira-Leggett model.
\end{abstract}
\maketitle
\section{Introduction.}
The relation between chaos and the phenomena of quantum dissipation
and decoherence has attracted a lot of attention in the last ten
years
\cite{cohen1,cohen2,cohen3,wilkinson,berry1,jarzynski,tulio,zurek1,zurek2,zurek3,cohen4,paz}.
The problem considered in most works involves the weak interaction
of a chaotic system with an external oscillator. Various points of
view have been considered by different authors. One approach
concentrates on the chaotic system itself, focusing on the
dissipation and treating the external oscillator as a time dependent
parameter that perturbs the chaotic system
\cite{cohen1,cohen2,cohen3}. The basic assumption is that the
external system is slow and sufficiently heavy not to be affected
significantly by the coupling
\cite{wilkinson,berry1,jarzynski,tulio}. The effect on the chaotic
system, on the other hand, is that of an adiabatic perturbation.
Under these conditions, a dissipative force, acting on the external
system, may result.
A different view of the same problem focus on the semiclassical
limit of chaotic systems. It has been shown
\cite{zurek1,zurek2,zurek3,cohen4,paz} that the coupling of a
chaotic system with an external environment, represented implicitly
by a small diffusion constant in the classical and quantum versions
of the Focker-Planck equation, leads to a very close correspondence
between the classical and quantum evolutions. The coupling causes
destruction of quantum interference and, at the same time, it washes
out the fine structures of the classical distributions, bringing the
two dynamics together.
In this paper we consider the chaotic system and the external
oscillator explicitly, as a single globally conservative Hamiltonian
system. We use the Feynman-Vernon approach to trace out the chaotic
system variables and construct an effective dynamics for the
oscillator, in close analogy with the treatment of the Brownian
motion considered by Caldeira and Leggett
\cite{caldeira1,caldeira2}. While we focus on the oscillator,
looking at dissipation and decoherence, the effects of the coupling
on the chaotic system are also taken into account consistently. This
characteristic of the Feynman-Vernon method turns out to be very
important in this problem, since both the oscillator and the chaotic
system have small number of degrees of freedom and are both affected
by the mutual interaction. However, whereas the treatment of
Caldeira and Leggett is phenomenological, in the sense that the
spectral properties of the reservoir are not derived from its
Hamiltonian, the case of a small chaotic environment has to be
treated dynamically. In other words, dissipation and decoherence
have to come out directly from correlations and response functions.
Our purpose here is to understand under what conditions a chaotic
system with only two degrees of freedom can produce dissipation and
decoherence, phenomena usually related to many body thermal baths.
In a previous paper \cite{bonanca} we have considered the
interaction of an oscillator with a chaotic system from a classical
point of view. We showed that the effects of the oscillator on the
environment cannot be neglected. Here we consider the quantum
version of the same problem, assuming the chaotic system to be in
the semiclassical regime.
In treatment of the Brownian motion by Caldeira and Leggett, the
degrees of freedom playing the role of the environment are averaged
with a canonical ensemble, since the reservoir is kept at the
constant temperature. Here, since the environment is small, the
microcanonical ensemble is more adequate. A similar approach was
recently considered by Esposito and Gaspard using random matrix
theory to model the chaotic environment
\cite{esposito1,esposito2,esposito3}.
The paper is organized as follows: in Sec.II, we review some aspects
of the classical formulation that are useful for the quantum
analysis. In Sec.III, we present the quantum formulation using
Feynman-Vernon approach \cite{feynman1}. The formal development
leads to quantum correlation and response functions, that we
calculate semiclassically. In Sec.IV, we consider two basic
applications: first, the propagation of an Gaussian state, where we
characterize quantum dissipation. Second, we calculate the evolution
of a superposition of two Gaussian states, focusing on the
decoherence due to the chaotic environment. In section V we present
our conclusions.
| 1,395 | 28,405 |
en
|
train
|
0.44.1
|
\section{Classical Formulation}
In this section we describe the behavior of a system of interest
coupled to a small chaotic environment from the classical point of
view. Although the formalism presented here can be extended to more
general systems, we particularize right away to the case of a
harmonic oscillator interacting with the so called Nelson system.
The discussion outlined here is a summary of the detailed results
presented in ref.\cite{bonanca} (see also
\cite{wilkinson,berry1,jarzynski,tulio}). The Hamiltonian of the
system is given by
\begin{eqnarray}
H = H_o(z) + H_c(x,y) + V_I(x,z), \label{eq1}
\end{eqnarray}
where
\begin{equation}
H_o(z) = \frac{p_z^2}{2m} + \frac{m\omega_0^2 z^2}{2},
\end{equation}
represents the system of interest,
\begin{equation}
V_I(x,z) = \gamma xz,
\end{equation}
is the interaction potential and
\begin{equation}
H_c(x,y) = \frac{p_x^2}{2} + \frac{p_y^2}{2} + \left(y - \frac{x^2}
{2}\right)^2 + 0.1\frac{x^2}{2}
\end{equation}
is the chaotic Hamiltonian, known as Nelson system (NS)
\cite{baranger}. The NS exhibits soft chaos and is fairly regular
for $E_c \,_{\sim}^<\,0.05$, strongly chaotic for $E_c \,_{\sim}^>\,
0.3$ and mixed for intermediate values of the energy.
In order to investigate the situation where $H_c$ plays the role of
an external environment for the oscillator, we assume that detailed
information about the chaotic system is not available. If the
environment were modeled by a heat bath, the only macroscopic
relevant information would be its temperature. In the present case
we assume that the only information available is the initial energy
$E_c(0)$ of the chaotic system. For the oscillator this implies that
only averages of its observables (over the chaotic system variables)
are accessible.
When the coupling between the chaotic system and the oscillator is
turned on, the overall conserved energy flows from one system to the
other. The oscillator's energy, in particular, fluctuates as a
function of the time for each specific trajectory. The oscillator's
average energy is calculated by taking an ensemble of initial
conditions uniformly distributed over the chaotic energy surface
$E_c(0)$. For the oscillator we fix only one initial condition,
which we choose to be $z(0) = 0$ and $p_z(0) = \sqrt{2mE_o (0)}$.
The microcanonical ensemble of chaotic initial conditions plus the
fixed oscillator's initial condition are propagated numerically and,
at each instant, $H_o$ is calculated for each trajectory and its
average value is computed. We have shown in \cite{bonanca} that the
oscillator's average energy $\langle E_o (t)\rangle$ tends to a
constant value for long times, indicating a 'thermalization' of the
systems.
The short time behavior of $\langle E_o(t)\rangle$ can be obtained
from the Linear Response Theory \cite{kubo}. From the equations of
motion for $z$ and $p_z$ we find
\begin{eqnarray}
z(t)=z_d (t)-\frac{\gamma}{m}\int_0^t
\mathrm{d}s\Gamma(t-s)x(s),\label{eq3} \\
p_z (t) = p_{z_d} (t)-\gamma\int_0^t \mathrm{d}s\chi(t-s)x(s), \label{eq4}
\end{eqnarray}
where $z_d (t)$ and $p_{z_d} (t)$ are the decoupled solutions, given
by,
\begin{equation}
z_d (t)=\frac {p_z (0)}{m\omega_0}\sin{(\omega_0t)} \qquad p_{z_d}
(t)=p_z (0)\cos{(\omega_0t)}\;,
\end{equation}
where $\Gamma(t-s) = \sin{[\omega_0(t-s)]}/\omega_0$ and $\chi(t-s)
= \cos{[\omega_0 (t-s)]}$.
Thus,
\begin{eqnarray}
\label{eq5}
\langle p_z^2 (t)\rangle = p^2_{z_d}(t)-2\gamma
p_{z_d} (t)\int_0^t
\mathrm{d}s\chi(t-s)\langle x(s)\rangle+\nonumber \\
\gamma^2\int_0^t\mathrm{d}s\int_0^t\mathrm{d}u
\chi(t-s)\chi(t-u)\langle x(s)x(u)\rangle
\end{eqnarray}
and
\begin{eqnarray}
\label{eq6} \langle z^2 (t)\rangle=z^2_{d}(t)-\frac{2\gamma}{m}
z_d (t)\int_0^t
\mathrm{d}s\Gamma(t-s)\langle x(s)\rangle+\nonumber \\
\frac{\gamma^2}{m^2}\int_0^t\mathrm{d}s\int_0^t\mathrm{d}u \Gamma(t-s)
\Gamma(t-u)\langle x(s)x(u)\rangle.
\end{eqnarray}
The oscillator's average energy can now be obtained from
\begin{equation}
\langle E_o (t)\rangle=\frac{\langle p_z^2 (t)\rangle}{2m}+\frac{m
\omega_0^2 \langle z^2 (t)\rangle}{2} \label{eq2}
\end{equation}
Equations (\ref{eq5}) and (\ref{eq6}) show that we need $\langle
x(t)\rangle$ and $\langle x(0)x(t)\rangle$ in order to calculate
$\langle E_o(t)\rangle$. The calculation of such averages involve
the distribution function $\rho(q,p;t)$ whose initial value is
$\rho(q,p;0) = \delta(H_c(q,p)-E_c(0))/\Sigma(E_c(0))$, with
$\Sigma(E_c(0))=\int \mathrm{d}q\mathrm{d}p\, \delta
(H_c(q,p)-E_c(0))$. Here we are using $q=(x,y)$ and $p=(p_x,p_y)$
for the coordinates and momenta of the chaotic system. If the
chaotic system were isolated, $\rho$ would be an invariant
distribution and $\rho(q,p;t)=\rho(q,p;0)$. The coupling, however,
causes the value of $H_c(q(t),p(t))$ to fluctuate in time,
distorting the energy surface $H_c=E_c(0)$. Linear Response Theory
provides the first order corrections to this distribution in the
limit of weak coupling \cite{kubo}. Keeping in (\ref{eq5}) and
(\ref{eq6}) only terms up to order $\gamma^2$, we find
\begin{eqnarray}
\langle x(t)\rangle=\langle x(t)\rangle_e -
\gamma\int_0^t\mathrm{d}s \phi_{xx} (t-s)z(s) \label{eq7}
\end{eqnarray}
and
\begin{eqnarray}
\langle x(0)x(t)\rangle=\langle x(0)x(t)\rangle_e, \label{eq8}
\end{eqnarray}
where $\langle A(q,p)\rangle_e=\int\mathrm{d}q\mathrm{d}p\,A(q,p)
\rho_e(q,p)$ and $\rho_e=\delta(H_c-E_c(0))/\Sigma(E_c(0))$ is the
microcanonical distribution of the isolated chaotic system.
$\phi_{xx} (t)$ is the response function, given by \cite{kubo}
\begin{eqnarray}
\phi_{xx}(t)=\langle \{x(0),x(t)\}\rangle_e=\int\int\mathrm{d}V\,
\rho_e\,\{x(0),x(t)\} \label{eq9}
\end{eqnarray}
where $\mathrm{d}V = \mathrm{d}x(0) \mathrm{d}y(0) \mathrm{d}
p_x(0)\mathrm{d}p_y(0)$ and $\{.,.\}$ is the Poisson bracket with
respect to the initial conditions. Since $H_c(-x)=H_c(x)$, $\langle
x(t)\rangle_e=0$. Substituting (\ref{eq7}) and (\ref{eq8}) into
(\ref{eq5}) and (\ref{eq6}) we obtain
\begin{equation}
\begin{array}{ll}
\langle p^2_z (t)\rangle &=
p^2_{z_d}(t)+2\gamma^2 p_{z_d}(t)\int_0^t\mathrm{d}s\chi(t-s)
\int_0^s\mathrm{d}u\phi_{xx}(s-u)z_d(u) \\
& + \gamma^2\int_0^t \mathrm{d}s\int_0^t
\mathrm{d}u\chi(t-s)\chi(t-u) \langle x(s)x(u)\rangle_e,
\label{eq10}
\end{array}
\end{equation}
\begin{equation}
\begin{array}{ll}
\langle z^2(t)\rangle & =
z^2_d(t)+\frac{2\gamma^2}{m}z_d(t)\int_0^t\mathrm{d}s
\Gamma(t-s)\int_0^s\mathrm{d}u\phi_{xx}(s-u)z_d(u) \\
&+\frac{\gamma^2}{m^2}\int_0^t \mathrm{d}s \int_0^t
\mathrm{d}u\Gamma(t-s)\Gamma(t-u)\langle x(s)x(u) \rangle_e.
\label{eq11}
\end{array}
\end{equation}
Eqs. (\ref{eq10}) and (\ref{eq11}) show that all the influence of
the chaotic system is contained in the functions $\langle
x(0)x(t)\rangle_e$ and $\phi_{xx}(t)$. For NS, the response function
is given by \cite{bonanca}
\begin{equation}
\phi_{xx}(t)=\frac{2}{E_c(0)}\langle p_x(0)x(t)\rangle_e.
\label{eq-resp}
\end{equation}
The correlation functions $\langle p_x(0)x(t)\rangle_e$ and $\langle
x(0)x(t)\rangle_e$ are obtained numerically with a fixed time step
symplectic integration algorithm \cite{forest} applied to the
isolated chaotic system. Fig. 1 shows the numerical correlation
functions for $E_c(0)=0.38$. These numerical results can be well
fitted by the expressions
\begin{equation}
\begin{array}{ll}
\langle x(0)x(t)\rangle_e &=\sigma e^{-\alpha t} \cos{\omega t},\\
\langle p_x (0) x(t)\rangle_e &=\mu e^{-\beta t} \sin{\Omega t},
\label{eq12}
\end{array}
\end{equation}
with decay rates $\alpha=0.0418$ and $\beta=0.0456$, amplitudes
$\sigma=1.865$ and $\mu=0.409$ and frequencies of oscillation
$\omega=0.1963$ and $\Omega=0.2043$ with $\chi^2\,\sim\,10^{-4}$
(see Fig.1). Notice that the exponents $\alpha$ and $\beta$ and the
frequencies $\omega$ and $\Omega$ are very similar. If the
interacting system were integrable, these functions would exhibit
quasi-periodic oscillations.
Considering the expressions (\ref{eq12}) and assuming
$\Omega\approx\omega$ and $\beta\approx\alpha$, we obtain the
following result for $\langle E_o(t)\rangle$
\begin{eqnarray}
\langle E_o(t)\rangle = E_o(0)+\frac{\gamma^2}{m}(B+At+f(t)+g(t)),
\label{eq13}
\end{eqnarray}
where $B$ is a constant, $f(t)$ is an oscillatory function and
$g(t)$ is proportional to $e^{-\alpha t}$. The important result is
the coefficient $A$
\begin{equation}
A=4\mu\omega\alpha \frac{\left[\frac{\sigma}{4\mu\omega}
(\omega^2_0+\omega^2+\alpha^2)-\frac{E_o(0)}
{E_c(0)}\right]}{[(\omega_0-\omega)^2+\alpha^2][(\omega_0+
\omega)^2+\alpha^2]}. \label{eq14}
\end{equation}
For fixed oscillator frequency $\omega_0$ and a given chaotic
energy shell $E_c(0)$ (and, consequently, for given $\sigma$,
$\mu$, $\omega$ and $\alpha$), the ratio $E_o(0)/E_c(0)$ is the
responsible for the average increase or decrease of $\langle
E_o(t)\rangle$. The short time equilibrium in the energy flow is
given by the condition $A=0$, or
\begin{equation}
\frac{E_o(0)}{E_c(0)}=\frac{\sigma}{4\mu\omega}(\omega^2_0+\omega^2
+\alpha^2) \;. \label{eq15}
\end{equation}
The equation of motion of $z(t)$ under the average effect of the
interaction with the chaotic system can also be written in terms of
the response function as
\begin{equation}
\langle\ddot{z}(t)\rangle+\omega^2_0 \langle z(t)\rangle
=-\frac{\gamma}{m}\langle x(t)\rangle
=\frac{\gamma^2}{m}\int_0^t\mathrm{d}s \, \phi_{xx} (t-s)
\langle z(s)\rangle.
\end{equation}
Integrating by parts yields
\begin{eqnarray}
\langle\ddot{z}(t)\rangle+\left(\omega^2_0-\frac{\gamma^2
F(0)}{m}\right) \langle z(t)\rangle+
\frac{\gamma^2}{m}\int_0^t\mathrm{d}s \, F(t-s) \langle
\dot{z}(s)\rangle+ \frac{\gamma^2}{m}z(0)F(t)=0, \label{eq16}
\end{eqnarray}
where
\begin{equation}
F(t-s) =\int\mathrm{d}s\,\phi_{xx}(t-s)=\frac{2\mu
e^{-\alpha(t-s)}}{E_c(0)(\alpha^2+\omega^2)}\{\omega
\cos{[\omega(t-s)]} +\alpha\sin{[\omega(t-s)]}\}. \label{eq17}
\end{equation}
Eq.(\ref{eq16}) shows that the interaction produces a harmonic
correction to the original potential, a dissipative term with memory
and an external force proportional to $z(0)$. The choice $z(0)=0$
simplifies (\ref{eq16}) and turns it into an average Langevin
equation.
Fig.2 shows a comparison between the numerically calculated `bare'
oscillator energy $\langle E_o(t)\rangle$, where
| 4,012 | 28,405 |
en
|
train
|
0.44.2
|
Eqs. (\ref{eq10}) and (\ref{eq11}) show that all the influence of
the chaotic system is contained in the functions $\langle
x(0)x(t)\rangle_e$ and $\phi_{xx}(t)$. For NS, the response function
is given by \cite{bonanca}
\begin{equation}
\phi_{xx}(t)=\frac{2}{E_c(0)}\langle p_x(0)x(t)\rangle_e.
\label{eq-resp}
\end{equation}
The correlation functions $\langle p_x(0)x(t)\rangle_e$ and $\langle
x(0)x(t)\rangle_e$ are obtained numerically with a fixed time step
symplectic integration algorithm \cite{forest} applied to the
isolated chaotic system. Fig. 1 shows the numerical correlation
functions for $E_c(0)=0.38$. These numerical results can be well
fitted by the expressions
\begin{equation}
\begin{array}{ll}
\langle x(0)x(t)\rangle_e &=\sigma e^{-\alpha t} \cos{\omega t},\\
\langle p_x (0) x(t)\rangle_e &=\mu e^{-\beta t} \sin{\Omega t},
\label{eq12}
\end{array}
\end{equation}
with decay rates $\alpha=0.0418$ and $\beta=0.0456$, amplitudes
$\sigma=1.865$ and $\mu=0.409$ and frequencies of oscillation
$\omega=0.1963$ and $\Omega=0.2043$ with $\chi^2\,\sim\,10^{-4}$
(see Fig.1). Notice that the exponents $\alpha$ and $\beta$ and the
frequencies $\omega$ and $\Omega$ are very similar. If the
interacting system were integrable, these functions would exhibit
quasi-periodic oscillations.
Considering the expressions (\ref{eq12}) and assuming
$\Omega\approx\omega$ and $\beta\approx\alpha$, we obtain the
following result for $\langle E_o(t)\rangle$
\begin{eqnarray}
\langle E_o(t)\rangle = E_o(0)+\frac{\gamma^2}{m}(B+At+f(t)+g(t)),
\label{eq13}
\end{eqnarray}
where $B$ is a constant, $f(t)$ is an oscillatory function and
$g(t)$ is proportional to $e^{-\alpha t}$. The important result is
the coefficient $A$
\begin{equation}
A=4\mu\omega\alpha \frac{\left[\frac{\sigma}{4\mu\omega}
(\omega^2_0+\omega^2+\alpha^2)-\frac{E_o(0)}
{E_c(0)}\right]}{[(\omega_0-\omega)^2+\alpha^2][(\omega_0+
\omega)^2+\alpha^2]}. \label{eq14}
\end{equation}
For fixed oscillator frequency $\omega_0$ and a given chaotic
energy shell $E_c(0)$ (and, consequently, for given $\sigma$,
$\mu$, $\omega$ and $\alpha$), the ratio $E_o(0)/E_c(0)$ is the
responsible for the average increase or decrease of $\langle
E_o(t)\rangle$. The short time equilibrium in the energy flow is
given by the condition $A=0$, or
\begin{equation}
\frac{E_o(0)}{E_c(0)}=\frac{\sigma}{4\mu\omega}(\omega^2_0+\omega^2
+\alpha^2) \;. \label{eq15}
\end{equation}
The equation of motion of $z(t)$ under the average effect of the
interaction with the chaotic system can also be written in terms of
the response function as
\begin{equation}
\langle\ddot{z}(t)\rangle+\omega^2_0 \langle z(t)\rangle
=-\frac{\gamma}{m}\langle x(t)\rangle
=\frac{\gamma^2}{m}\int_0^t\mathrm{d}s \, \phi_{xx} (t-s)
\langle z(s)\rangle.
\end{equation}
Integrating by parts yields
\begin{eqnarray}
\langle\ddot{z}(t)\rangle+\left(\omega^2_0-\frac{\gamma^2
F(0)}{m}\right) \langle z(t)\rangle+
\frac{\gamma^2}{m}\int_0^t\mathrm{d}s \, F(t-s) \langle
\dot{z}(s)\rangle+ \frac{\gamma^2}{m}z(0)F(t)=0, \label{eq16}
\end{eqnarray}
where
\begin{equation}
F(t-s) =\int\mathrm{d}s\,\phi_{xx}(t-s)=\frac{2\mu
e^{-\alpha(t-s)}}{E_c(0)(\alpha^2+\omega^2)}\{\omega
\cos{[\omega(t-s)]} +\alpha\sin{[\omega(t-s)]}\}. \label{eq17}
\end{equation}
Eq.(\ref{eq16}) shows that the interaction produces a harmonic
correction to the original potential, a dissipative term with memory
and an external force proportional to $z(0)$. The choice $z(0)=0$
simplifies (\ref{eq16}) and turns it into an average Langevin
equation.
Fig.2 shows a comparison between the numerically calculated `bare'
oscillator energy $\langle E_o(t)\rangle$, where
\begin{equation}
E_o(z) = \frac{p_z^2}{2m} + \frac{m\omega_0^2 z^2}{2},
\end{equation}
the `re-normalized' oscillator energy $\langle E_{or}(t)\rangle$,
where
\begin{equation}
E_{or}(z) = \frac{p_z^2}{2m} + \frac{m\omega_0^2 z^2}{2}
-\frac{\gamma^2}{2}F(0)z^2, \label{eq18}
\end{equation}
and the expression (\ref{eq13}) without the oscillating term $f(t)$.
We have chosen $\gamma$ and $m$ so that $\omega^2_0 - \gamma^2
F(0)/m>0$. We also have chosen $\omega_0$ so that
$e^{-\alpha/\omega_0}\approx 10^{-4}$ and $g(t)$ decreases very
fast. In this case only the linear and the oscillating terms in
Eq.(\ref{eq13}) are important. We have subtracted the oscillating
part of Eq.(\ref{eq13}) in Fig.2 to highlight the linear increase or
decrease in the average energy. In the time scale of Fig.2, which
corresponds to several periods of the decoupled oscillator, the
linear behavior describes very well the numerical results. Fig.2(b)
shows the equilibrium situation according to Eq.(\ref{eq15}). Notice
that $F(t)$ decays very fast in the time scale $1/\omega_0$, leading
to the dissipative force in Eq.(\ref{eq16}).
In the next sections we consider the quantum counterpart of these
classical calculations. The chaotic system will be treated
semiclassically and the quantum versions of the response and
correlation functions will play important roles.
| 1,846 | 28,405 |
en
|
train
|
0.44.3
|
\section{Quantum Formulation}
| 8 | 28,405 |
en
|
train
|
0.44.4
|
\subsection{The Feynman-Vernon Approach}
In this section we describe the dynamics of the coupled oscillator
from a quantum point of view. In order to do that we need, like in
the classical case, a systematic way to eliminate the detailed
information we don't need about the chaotic system. We will do that
using the Feynman-Vernon approach \cite{feynman1}. Because of the
non-linear chaotic system, we will not be able to perform exact
calculations. Instead, we will resort to semiclassical
approximations.
We consider the quantum version of the full Hamiltonian,
Eq.(\ref{eq1}), and again we denote by $q=(x,y)$ the pair of
coordinates of the chaotic system. The density matrix operator can
be written as
\begin{eqnarray}
\hat{\rho}(T)=|\psi(T)\rangle \langle\psi(T)| = e^{-i\hat{H}T/\hbar}
|\psi(0)\rangle \langle\psi(0)|e^{i\hat{H}T/\hbar} \nonumber
\end{eqnarray}
where $\psi(T)$ is the wave function of the whole system. In the
position representation
\begin{eqnarray}
\lefteqn{\rho(z(T),q(T),z'(T),q'(T))=\langle z,q|\psi(T)\rangle
\langle\psi(T)|z',q'\rangle} \nonumber \\
&=&\int\mathrm{d}z(0)\mathrm{d}z'(0)\mathrm{d}q(0)\mathrm{d}q'(0)
\langle z,q|e^{-i\hat{H}T/\hbar}|z(0),q(0)\rangle\langle z(0),q(0)|
\psi(0)\rangle \times \nonumber \\
& &\langle\psi(0)|z'(0),q'(0)\rangle\langle z'(0),q'(0)|
e^{i\hat{H}T/\hbar}|z',q'\rangle \nonumber \\
&=&\int\mathrm{d}z(0)\mathrm{d}z'(0)\mathrm{d}q(0)\mathrm{d}q'(0)
K(z(T),q(T),z(0),q(0))\psi(z(0),q(0)) \times \nonumber \\
& &K^*(z'(T),q'(T),z'(0),q'(0))\psi^*(z'(0),q'(0)),
\end{eqnarray}
where the propagators can be written in terms of Feynman path
integrals as \cite{feynman2}
\begin{eqnarray}
K(z(T),q(T),z(0),q(0))=\int\mathrm{D}z(t)\mathrm{D}q(t)\exp{\left[
\frac{i}{\hbar}S[z(t),q(t)]\right]},
\end{eqnarray}
with action
\begin{eqnarray}
S[z(t),q(t)]&=& \int_0^T\mathrm{d}t
(L_0(z(t))+L_c(q(t))+L_I(z(t),q(t))) \nonumber \\
&\equiv &S_o+S_c+S_I.
\end{eqnarray}
Thus,
\begin{eqnarray}
\lefteqn{\rho(z(T),q(T),z'(T),q'(T))=} \nonumber \\
&=&\int\mathrm{d}z(0)\mathrm{d}z'(0)
\mathrm{d}q(0)\mathrm{d}q'(0)\mathrm{D}z(t)\mathrm{D}z'(t)\mathrm{D}q(t)
\mathrm{D}q'(t)\nonumber \\
&
&\exp\left[\frac{i}{\hbar}\left(S[z(t),q(t)]-S[z'(t),q'(t)]\right)\right]
\rho(z(0),q(0),z'(0),q'(0)),
\end{eqnarray}
where
\begin{equation}
\rho(z(0),q(0),z'(0),q'(0))=\psi(z(0),q(0))\psi^*(z'(0),q'(0))
\end{equation}
is the initial state. As usual we use $z(T)$ and $q(T)$ just as
labels for the positions $z$ and $q$ at $T$. $z(T)$ and $q(T)$ are
not functions of $T$.
We assume that the initial state can be written as
\begin{equation}
\rho(z(0),q(0),z'(0),q'(0))=\rho_o(z(0),z'(0))\rho_c(q(0),q'(0))
\end{equation}
and define the reduced density matrix by
\begin{equation}
\rho_o(z(T),z'(T))=\int\mathrm{d}q(T)\rho(z(T),q(T),z'(T),q(T)).
\end{equation}
We obtain
\begin{eqnarray}
\lefteqn{\rho_o(z(T),z'(T))=\int\mathrm{d}z(0)\mathrm{d}z'(0)\mathrm{D}z(t)
\mathrm{D}z'(t)} \nonumber \\
& &\left\{\int\mathrm{d}q(0)\mathrm{d}q'(0)\mathrm{d} q(T)
\mathrm{d}q'(T)\mathrm{D}q(t)
\mathrm{D}q'(t)\delta(q(T)-q'(T))\right.
\nonumber \\
& &\exp{\left[\frac{i}{\hbar}(S_c[q(t)]-S_c[q'(t)]+S_I[z(t),
q(t)] -S_I[z'(t),q'(t)\right]} \nonumber\\
& &\rho_c(q(0),q'(0)) \bigg\} \exp{\left[\frac{i}
{\hbar}\left(S_o[z(t)]-S_o[z'(t)]\right)\right]\rho_o(z(0),z'(0))},
\end{eqnarray}
which can be written as
\begin{equation}
\rho_o(z(T),z'(T))=\int\mathrm{d}z(0)\mathrm{d}z'(0)J(z(T),z'(T),z(0),
z'(0))\rho_o(z(0),z'(0)), \label{eq19}
\end{equation}
where
\begin{equation}
J(z(T),z'(T),z(0),z'(0))=\int\mathrm{D}z(t)\mathrm{D}z'(t)\mathcal{F}
[z(t),z'(t)]\exp[\frac{i}{\hbar}(S_o[z(t)]-S_o[z'(t)])]\label{eq20}
\end{equation}
is the `superpropagator' and
\begin{eqnarray}
\lefteqn{\mathcal{F}[z(t),z'(t)]=\int\mathrm{d}q(0)\mathrm{d}q'(0)
\mathrm{d}q(T)
\mathrm{d}q'(T)\mathrm{D}q(t)\mathrm{D}q'(t)\delta(q(T)-q'(T))}
\nonumber \\ &
&\exp\left[\frac{i}{\hbar}\left(S_c[q(t)]-S_c[q'(t)]+S_I[z(t),q(t)]-
S_I[z'(t),q'(t)]\right)\right] \; \rho_c(q(0),q'(0)),
\end{eqnarray}
is the so called `influence functional', which contains all the
information about the chaotic system.
Equation (\ref{eq19}) is the equation of motion for the reduced
density matrix. For $\mathcal{F}=1$, $J$ becomes the propagator of
the isolated oscillator. Our goal is to get an approximate
expression for $J$ that includes the effects of the chaotic system.
We take as the initial state for the chaotic system one of its
energy eigenstate $\phi_a(q)=\langle q|a\rangle$. Thus,
\begin{equation}
\rho_c(q(0),q'(0))=\phi_a(q(0))\phi_a^*(q'(0)).
\end{equation}
This is the quantum version of the classical microcanonical
distribution we considered in section II.
The difficulty in the calculation of $J$ is that the chaotic
Lagrangian $L_c$ is not quadratic. Therefore, $\mathcal{F}$ has to
be treated in a perturbative manner. Rewriting $\mathcal{F}$ as
\begin{eqnarray}
\lefteqn{\mathcal{F}[z(t),z'(t)]=\int\mathrm{d}q(0)\ldots
\mathrm{D}q'(t)\delta(q(T)-
q'(T))\exp\left[\frac{i}{\hbar}(S_c[q(t)]-S_c[q'(t)])\right]}\nonumber\\
&
&\exp\left[-\gamma\frac{i}{\hbar}\left(\int_0^T\mathrm{d}t(z(t)x(t)-
z'(t)x'(t))\right)\right]\phi_a(q(0))\phi_a^*(q'(0))
\end{eqnarray}
we assume that $\gamma$ is small enough so that the exponential in
the second line can be expanded to second order in its argument.
These terms can be calculated by inserting complete sets of energy
eigenstates of $H_c$. The result, following \cite{feynman1}, is
\begin{eqnarray}
\lefteqn{\mathcal{F}[z(t),z'(t)] \approx
1-\left(\frac{i\gamma}{\hbar}\right)x_{aa}
\int_0^T\mathrm{d}t[z(t)-z'(t)]} \nonumber \\
& &-\left(\frac{\gamma^2}{\hbar}\right)
\int_0^T\mathrm{d}t\int_0^t\mathrm{d}s[z(t)-z'(t)][z(s)F_a^*(t-s)
-z'(s)F_a(t-s)],
\end{eqnarray}
where
\begin{eqnarray}
F_a(t-s) = \sum_b\frac{|x_{ba}|^2}{\hbar}\exp[i\omega_{ba}(t-s)],
\qquad\omega_{ba}=\frac{E_b-E_a}{\hbar},\label{eq21}
\end{eqnarray}
\begin{eqnarray}
x_{ba}=\int\mathrm{d}q\phi_b^*(q)x\phi_a(q),
\end{eqnarray}
and $E_b$ are the eigen-energies of chaotic system ($x$ is the
coordinate of $H_c$ in $V_I$ ). For the NS $H_c(-x)=H_c(x)$ and
$x_{aa}=0$. Thus,
\begin{eqnarray}
\mathcal{F}[z(t),z'(t)] \approx 1-\frac{1}{\hbar}\Phi[z(t),z'(t)]
\approx \exp\left[-\frac{1}{\hbar}\Phi[z(t),z'(t)]\right]
\end{eqnarray}
where
\begin{eqnarray}
\Phi[z,z'] = \frac{\gamma^2}{\hbar}
\int_0^T\mathrm{d}t\int_0^t\mathrm{d}s[z(t)-z'(t)][z(s)F_a^*(t-s)-
z'(s)F_a(t-s)].
\end{eqnarray}
With these approximations the superpropagator can be written as
\begin{eqnarray}
J(z(T),z'(T),z(0),z'(0))=\int\mathrm{D}z(t)\mathrm{D}z'(t) \, e^{
\frac{i}{\hbar}(\tilde{S}_{ef}[z(t),z'(t)])},
\end{eqnarray}
where we have defined the effective action
\begin{equation}
\tilde{S}_{ef}[z(t),z'(t)]=S_o[z(t)]-S_o[z'(t)]+i\Phi[z(t),z'(t)].
\label{sefect}
\end{equation}
Since $\tilde{S}_{ef}$ is quadratic in $z$ and $z'$ the path
integral can be solved exactly by the stationary phase method. It is
convenient to define the new variables $r(t)=(z(t)+z'(t))/2$ and
$y(t)=z(t)-z'(t)$ \cite{weiss} and to separate
$F_a(t)=F_a^{'}(t)+iF_a^{''}(t)$ into real and imaginary parts. This
allows us to write
\begin{eqnarray}
J(r(T),y(T),r(0),y(0))= \int\mathrm{D}r(t)\mathrm{D}y(t)\, e^{
\frac{i}{\hbar}\tilde{S}[r(t),y(t)] \, - \, \frac{1}{\hbar}
\phi[r(t),y(t)]} \label{pathint}.
\end{eqnarray}
where
\begin{eqnarray}
\tilde{S}[r(t),y(t)]\equiv\int_0^T\mathrm{d}t\left\{m[\dot{r}(t)\dot{y}(t)
-\omega_0^2r(t)y(t)]+2\gamma^2y(t)\int_0^t\mathrm{d}sF_a^{''}(t-s)
r(s)\right\}, \label{sefreal}
\end{eqnarray}
and
\begin{eqnarray}
\phi[r(t),y(t)]\equiv \gamma^2\int_0^T\mathrm{d}t\int_0^t\mathrm{d}s
y(t)y(s)F_a^{'}(t-s), \label{sefimag}
\end{eqnarray}
are the real and imaginary parts of $\tilde{S}_{ef}$. In Appendix A
we show that
\begin{eqnarray}
J(r(T),y(T),r(0),y(0))=G(T,0)\exp\left\{\frac{i}{\hbar}\tilde{S}[r_e,y_e]
\right\}\exp\left\{-\frac{1}{\hbar}\phi[y_e,y_e]\right\},
\label{superprop}
\end{eqnarray}
where $G(T,0)$ can be obtained by the normalization condition of
reduced density matrix and $r_e(t)$ and $y_e(t)$ are the extremum
paths of $\tilde{S}$, which satisfy
\begin{eqnarray}
\ddot{r}_e(t)+
\omega_0^2r_e(t)-\frac{2\gamma^2}{m}\int_0^t\mathrm{d}sF_a{''}(t-s)
r_e(s)=0,\label{eq22} \\
\ddot{y}_e(t)+
\omega_0^2y_e(t)-\frac{2\gamma^2}{m}\int_0^t\mathrm{d}sF_a{''}(s-t)
y_e(s)=0.\label{eq23}
\end{eqnarray}
Therefore we need $F_a^{''}$ to solve (\ref{eq22}) and (\ref{eq23})
and we also need $F_a^{'}$ to calculate $\phi[y_e,y_e]$.
From Eq.(\ref{eq21}) it follows that
\begin{eqnarray}
F_a(t)=\frac{\langle
a|\hat{x}(0)\hat{x}(t)|a\rangle}{\hbar},\label{eq24}
\end{eqnarray}
where $\hat{x}(t)$ is the Heisenberg representation of $\hat{x}$.
The real and imaginary parts of $F_a$ are
| 3,999 | 28,405 |
en
|
train
|
0.44.5
|
where
\begin{eqnarray}
F_a(t-s) = \sum_b\frac{|x_{ba}|^2}{\hbar}\exp[i\omega_{ba}(t-s)],
\qquad\omega_{ba}=\frac{E_b-E_a}{\hbar},\label{eq21}
\end{eqnarray}
\begin{eqnarray}
x_{ba}=\int\mathrm{d}q\phi_b^*(q)x\phi_a(q),
\end{eqnarray}
and $E_b$ are the eigen-energies of chaotic system ($x$ is the
coordinate of $H_c$ in $V_I$ ). For the NS $H_c(-x)=H_c(x)$ and
$x_{aa}=0$. Thus,
\begin{eqnarray}
\mathcal{F}[z(t),z'(t)] \approx 1-\frac{1}{\hbar}\Phi[z(t),z'(t)]
\approx \exp\left[-\frac{1}{\hbar}\Phi[z(t),z'(t)]\right]
\end{eqnarray}
where
\begin{eqnarray}
\Phi[z,z'] = \frac{\gamma^2}{\hbar}
\int_0^T\mathrm{d}t\int_0^t\mathrm{d}s[z(t)-z'(t)][z(s)F_a^*(t-s)-
z'(s)F_a(t-s)].
\end{eqnarray}
With these approximations the superpropagator can be written as
\begin{eqnarray}
J(z(T),z'(T),z(0),z'(0))=\int\mathrm{D}z(t)\mathrm{D}z'(t) \, e^{
\frac{i}{\hbar}(\tilde{S}_{ef}[z(t),z'(t)])},
\end{eqnarray}
where we have defined the effective action
\begin{equation}
\tilde{S}_{ef}[z(t),z'(t)]=S_o[z(t)]-S_o[z'(t)]+i\Phi[z(t),z'(t)].
\label{sefect}
\end{equation}
Since $\tilde{S}_{ef}$ is quadratic in $z$ and $z'$ the path
integral can be solved exactly by the stationary phase method. It is
convenient to define the new variables $r(t)=(z(t)+z'(t))/2$ and
$y(t)=z(t)-z'(t)$ \cite{weiss} and to separate
$F_a(t)=F_a^{'}(t)+iF_a^{''}(t)$ into real and imaginary parts. This
allows us to write
\begin{eqnarray}
J(r(T),y(T),r(0),y(0))= \int\mathrm{D}r(t)\mathrm{D}y(t)\, e^{
\frac{i}{\hbar}\tilde{S}[r(t),y(t)] \, - \, \frac{1}{\hbar}
\phi[r(t),y(t)]} \label{pathint}.
\end{eqnarray}
where
\begin{eqnarray}
\tilde{S}[r(t),y(t)]\equiv\int_0^T\mathrm{d}t\left\{m[\dot{r}(t)\dot{y}(t)
-\omega_0^2r(t)y(t)]+2\gamma^2y(t)\int_0^t\mathrm{d}sF_a^{''}(t-s)
r(s)\right\}, \label{sefreal}
\end{eqnarray}
and
\begin{eqnarray}
\phi[r(t),y(t)]\equiv \gamma^2\int_0^T\mathrm{d}t\int_0^t\mathrm{d}s
y(t)y(s)F_a^{'}(t-s), \label{sefimag}
\end{eqnarray}
are the real and imaginary parts of $\tilde{S}_{ef}$. In Appendix A
we show that
\begin{eqnarray}
J(r(T),y(T),r(0),y(0))=G(T,0)\exp\left\{\frac{i}{\hbar}\tilde{S}[r_e,y_e]
\right\}\exp\left\{-\frac{1}{\hbar}\phi[y_e,y_e]\right\},
\label{superprop}
\end{eqnarray}
where $G(T,0)$ can be obtained by the normalization condition of
reduced density matrix and $r_e(t)$ and $y_e(t)$ are the extremum
paths of $\tilde{S}$, which satisfy
\begin{eqnarray}
\ddot{r}_e(t)+
\omega_0^2r_e(t)-\frac{2\gamma^2}{m}\int_0^t\mathrm{d}sF_a{''}(t-s)
r_e(s)=0,\label{eq22} \\
\ddot{y}_e(t)+
\omega_0^2y_e(t)-\frac{2\gamma^2}{m}\int_0^t\mathrm{d}sF_a{''}(s-t)
y_e(s)=0.\label{eq23}
\end{eqnarray}
Therefore we need $F_a^{''}$ to solve (\ref{eq22}) and (\ref{eq23})
and we also need $F_a^{'}$ to calculate $\phi[y_e,y_e]$.
From Eq.(\ref{eq21}) it follows that
\begin{eqnarray}
F_a(t)=\frac{\langle
a|\hat{x}(0)\hat{x}(t)|a\rangle}{\hbar},\label{eq24}
\end{eqnarray}
where $\hat{x}(t)$ is the Heisenberg representation of $\hat{x}$.
The real and imaginary parts of $F_a$ are
\begin{eqnarray}
F_a^{'}(t)=\frac{\langle a| \{\hat{x}(0),\hat{x}(t)\}| a
\rangle}{2\hbar},
\end{eqnarray}
and
\begin{eqnarray}
F_a^{''}(t)=\frac{\langle
a|[\hat{x}(0),\hat{x}(t)]|a\rangle}{2i\hbar},
\end{eqnarray}
where $\{.\}$ is the anticomutator and $[.]$ is the comutator. Thus,
$F_a^{'}$ and $F_a^{''}$ are, respectively, the quantum analogs of
the classical correlation and response functions of Section II.
| 1,616 | 28,405 |
en
|
train
|
0.44.6
|
\subsection{Semiclassical Expressions for Correlation Functions.}
In this section we obtain semiclassical formulas for $F_a^{'}(t)$
and $F_a^{''}(t)$. We write
\begin{eqnarray}
F_a^{'}(t) = \frac{1}{2\hbar} \sum_b\langle
b|\{\hat{x}(0),\hat{x}(t)\}\hat{\rho}|b\rangle =
\frac{1}{2\hbar}Tr(\hat{f}(t)\hat{\rho}),\label{eq25}
\end{eqnarray}
where $\hat{\rho}=|a\rangle\langle a|$ is the microcanonical
distribution of the chaotic system and $\hat{f} =
\{\hat{x}(0),\hat{x}(t)\}$. To calculate $F''_a$ we take
$\hat{f}=-i[\hat{x}(0),\hat{x}(t)]$. Using the Wigner-Weyl
representation \cite{wigner} the trace can be written as
\begin{eqnarray}
Tr[\hat{f}(t)\hat{\rho}]=\int\frac{\mathrm{d}^2 q\mathrm{d}^2 p}
{(2\pi\hbar)^2}f(q,p;t)W(q,p) \label{eq26}
\end{eqnarray}
where
\begin{eqnarray}
f(q,p;t)=\int_{-\infty}^{\infty}\mathrm{d}u \, e^{
\frac{i}{\hbar}p\cdot u}\langle
q-u/2|\hat{f}(t)|q+u/2\rangle.\label{eq27}
\end{eqnarray}
is the Weyl transformation (or symbol) of $\hat{f}(t)$ and $W(q,p)$
is the Wigner function of $\hat{\rho}$. For
$\hat{f}(t)=-i[\hat{x}(0),\hat{x}(t)]$ we have
\begin{eqnarray}
f(q,p;t)&=&-i\int_{-\infty}^{\infty}\mathrm{d}u \, e^{ ip\cdot
u/\hbar} \, \langle
q-u/2|(\hat{x}\hat{x}(t)-\hat{x}(t)\hat{x})|q+u/2\rangle
\nonumber \\
&=&-i\int_{-\infty}^{\infty}\mathrm{d}u \, e^{ ip\cdot u/\hbar} \,
(-u_x)\langle q-u/2|\hat{x}(t)|q+u/2\rangle
\nonumber \\
&=&\hbar\frac{\partial}{\partial
p_x}\left\{\int_{-\infty}^{\infty}\mathrm{d} u \, e^{ ip\cdot
u/\hbar} \,\langle
q-u/2|\hat{x}(t)|q+u/2\rangle\right\} \nonumber \\
&=& \hbar\frac{\partial}{\partial p_x} \left\{
\int_{-\infty}^{\infty}\mathrm{d} u\, e^{ ip\cdot u/\hbar} \,
\int_{-\infty}^{\infty}\mathrm{d} v\, v_x
K^*(v,q-u/2;t)K(v,q+u/2;t)\right\} \label{eq28}
\end{eqnarray}
where $u$ and $v$ represent coordinates of the chaotic system and
$K(v,v';t)=\langle v|e^{-i \hat{H}_c t/\hbar}|v'\rangle$ is the
propagator.
We now replace the propagators by their semiclassical expressions
$\tilde{K}$ and do the integrals by the stationary phase
approximation. The stationary phase condition shows that the most
important contributions come from the trajectories starting at
$q-u/2$ (for $K^*$) and $q+u/2$ (for $K$) and arriving at $v$ in the
time $t$ such that
\begin{eqnarray}
\nabla_{v}R_k(v,q-u/2)-\nabla_{v}R_l(v, q+u/2)=0,\label{eq30}
\end{eqnarray}
where $R_k$ and $R_l$ are Hamilton's principal functions coming from
the phases in $\tilde{K}$ and $\tilde{K}^*$. Since $\nabla_{v}
R_i(v,v')$ gives the final momentum, (\ref{eq30}) imposes that the
final momenta of the two trajectories must be equal. Since the final
positions are also equal, the two trajectories must be identical.
Thus,
\begin{eqnarray}
\int_{-\infty}^{\infty}\mathrm{d}v\, v_x \tilde{K}^*(
v,q-u/2;t)\tilde{K}( v,q+ u/2;t)\approx x(q,p;t)\delta(u),
\label{eq31}
\end{eqnarray}
where $x(q,p;t)$ is the coordinate $x$ of the stationary trajectory.
Using (\ref{eq31}) in (\ref{eq28}) we find
\begin{eqnarray}
f(q,p;t)=\hbar\frac{\partial}{\partial p_x} x (q,p;t) =
\hbar\{x(0),x(t)\}.\label{eq32}
\end{eqnarray}
since
\begin{eqnarray}
\{x(0),x(t)\}=\frac{\partial x(0)}{\partial x(0)}\frac{\partial
x(t)}{\partial p_x(0)}-\frac{\partial x(0)}{\partial
p_x(0)}\frac{\partial x(t)}{\partial x(0)}=\frac{\partial
x(t)}{\partial p_x(0)}.\label{eq33}
\end{eqnarray}
For $\hat{f}(t)=\{\hat{x}(0),\hat{x}(t)\}$, we find
\begin{eqnarray}
f(q,p;t)&=&
\int_{-\infty}^{\infty}\mathrm{d}u\exp\left(\frac{i}{\hbar} p\cdot
u\right)2q_x\langle q-u/2|\hat{X}(t)|q+u/2\rangle \nonumber
\\ &\approx& 2q_x\, x(q,p;t)=2x(0)x(t). \label{eq35}
\end{eqnarray}
The semiclassical limit of the Wigner function
\begin{eqnarray}
W(q,p;E)=\frac{1}{\hbar}\int_{-\infty}^{\infty}\mathrm{d}t
e^{iEt/\hbar}\int\mathrm{d}u\exp\left(\frac{i}{\hbar}p\cdot u
\right)K(q+u/2,q-u/2;t).\label{eq37}
\end{eqnarray}
was obtained by Berry \cite{berry2} and can be written as
\begin{eqnarray}
W(q,p;E)\approx\delta(E-H(q,p))+W_1(q,p;E). \label{eq38}
\end{eqnarray}
The first term is the classical micro-canonical distribution and the
second, $W_1$, is given by classical periodic orbits corrections to
the classical function. These periodic orbits have energy $E=E_a$,
corresponding to the eigenstate $|a\rangle$ of the microcanonical
quantum distribution. Using (\ref{eq38}) we write
\begin{eqnarray}
Tr[\hat{f}(t)\hat{\rho}] &\approx
\displaystyle{\frac{1}{(2\pi\hbar)^2}} & \left[ \int\mathrm{d}q
\mathrm{d} p f(q,p;t)\delta(E-H(q,p)) \, + \int\mathrm{d}q
\mathrm{d} p
f(q,p;t)W_1(q,p;E) \right]\nonumber \\
& & \equiv \frac{1}{(2\pi\hbar)^2} \left[f^0 + f^1\right].
\label{eq39}
\end{eqnarray}
When $f(q,p;t)$ is replaced by the semiclassical expressions for the
anticomutator and comutator, the first term of (\ref{eq39}) becomes,
except for a normalization, the classical expressions for the
response and correlation functions respectively. Following
\cite{berry2} the second term of (\ref{eq39}) becomes
\begin{eqnarray}
f^1(E,t) \approx \sum_jA_j\cos{(S_j(E)/\hbar+\gamma_j)}\oint
\mathrm{d}\tau f(q_j(\tau),p_j(\tau);t).\label{eq40}
\end{eqnarray}
where, $A_j$ depends on the stability of $j$-periodic orbit,
$S_j(E)$ is its action, $\gamma_j$ is the Maslov index and the
integral is calculated over a period of the $j$-orbit. Analogous
results can be found, for example, in \cite{eckart}.
Furthermore \cite{berry2},
\begin{equation}
\begin{array}{ll}
Tr(\hat{\rho})&=\displaystyle{\frac{1}{(2\pi\hbar)^2}} \left[
\int\mathrm{d}q\mathrm{d}p
\,\delta(E-H(q,p))+n_q(E;\hbar) \right] \\ \\
&=\displaystyle{\frac{1}{(2\pi\hbar)^2}}[n_c(E)+ n_q(E;\hbar)]\\
\label{eq42}
\end{array}
\end{equation}
where the first term is the classical density of states and the
second term is known as Gutzwiller's trace formula. Thus,
\begin{equation}
\begin{array}{ll}
\displaystyle{\frac{Tr[\hat{f}(t)\hat{\rho}]}{Tr[\hat{\rho}]}} &
\approx \displaystyle{\frac{f^0(E;t)+f^1(E;t)}{n_c(E)+n_q(E;\hbar)}}
\\ \\ & \approx \langle f(q,p;t)\rangle_{cl.}\left[1-
\frac{n_q(E;\hbar)}{n_c(E)}\right]+\frac{f^1(E;t)}{n_c(E)}\left[1-
\frac{n_q(E;\hbar)}{n_c(E)}\right] \label{eq47}
\end{array}
\end{equation}
where $\langle f(q,p;t)\rangle_{cl.}=f^0(E,t)/n_c(E)$.
Finally, we can calculate the semiclassical expressions for
$F_a^{''}(t)$ and $F_a^{'}(t)$. From (\ref{eq32}), (\ref{eq35}) and
(\ref{eq47})
\begin{eqnarray}
F_a^{'}(t)&\approx& \frac{\langle
x(0)x(t)\rangle_{cl.}}{\hbar}
\left[1-\frac{n_q(E;\hbar)}{n_c(E)}\right]
\nonumber \\
&+&\frac{1}{\hbar n_c(E)}\sum_j A_j
\cos\left(S_j(E)/\hbar+\gamma_j\right) \oint\mathrm{d}\tau
x_j(\tau)x_j(\tau+t).\label{eq49}
\end{eqnarray}
and
\begin{eqnarray}
F_a^{''}(t)=\frac{\langle\hat{O}(t)\rangle}{2\hbar}&\approx&\frac{\langle
\{x(0),x(t)\}\rangle_{cl.}}{2}
\left[1-\frac{n_q(E;\hbar)}{n_c(E)}\right]
\nonumber \\
&+&\frac{1}{2 n_c(E)}\sum_j A_j \cos\left(S_j(E)/\hbar+\gamma_j\right)
\oint\mathrm{d}\tau \{x_j(\tau),x_j(\tau+t)\}.\nonumber \\
\,\label{eq50}
\end{eqnarray}
Both these semiclassical expressions are given by their classical
counterparts multiplied by a correction to their amplitudes, given
by $n_q(E;\hbar)/n_c(E)$, plus a correction from periodic orbits.
The temporal dependence of the first is given solely by the
classical dynamics and it decays exponentially. The second term,
however, is a sum of oscillating functions and carries the temporal
dependence characteristic of the chaotic system. As a final remark
we note in \cite{esposito1} these functions were calculated using
random matrix theory.
| 3,124 | 28,405 |
en
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.