text
stringlengths 73
2.82k
| category
stringclasses 21
values |
|---|---|
Bose-Einstein Condensation with Optimal Rate for Trapped Bosons in the
Gross-Pitaevskii Regime: We consider a Bose gas consisting of $N$ particles in $\mathbb{R}^3$, trapped
by an external field and interacting through a two-body potential with
scattering length of order $N^{-1}$. We prove that low energy states exhibit
complete Bose-Einstein condensation with optimal rate, generalizing previous
work in \cite{BBCS1, BBCS4}, restricted to translation invariant systems. This
extends recent results in \cite{NNRT}, removing the smallness assumption on the
size of the scattering length. | math-ph |
Limit theorems for the cubic mean-field Ising model: We study a mean-field spin model with three- and two-body interactions. The
equilibrium measure for large volumes is shown to have three pure states, the
phases of the model. They include the two with opposite magnetization and an
unpolarized one with zero magnetization, merging at the critical point. We
prove that the central limit theorem holds for a suitably rescaled
magnetization, while its violation with the typical quartic behavior appears at
the critical point. | math-ph |
Mass scaling of the near-critical 2D Ising model using random currents: We examine the Ising model at its critical temperature with an external
magnetic field $h a^{\frac{15}{8}}$ on $a\mathbb{Z}^2$ for $a,h >0$. A new
proof of exponential decay of the truncated two-point correlation functions is
presented. It is proven that the mass (inverse correlation length) is of the
order of $h^\frac{8}{15}$ in the limit $h \to 0$. This was previously proven
with CLE-methods in $\lbrack 1 \rbrack$. Our new proof uses instead the random
current representation of the Ising model and its backbone exploration. The
method further relies on recent couplings to the random cluster model $\lbrack
2 \rbrack$ as well as a near-critical RSW-result for the random cluster model
$\lbrack 3 \rbrack$. | math-ph |
Algebraic (super-)integrability from commutants of subalgebras in
universal enveloping algebras: Starting from a purely algebraic procedure based on the commutant of a
subalgebra in the universal enveloping algebra of a given Lie algebra, the
notion of algebraic Hamiltonians and the constants of the motion generating a
polynomial symmetry algebra is proposed. The case of the special linear Lie
algebra $\mathfrak{sl}(n)$ is discussed in detail, where an explicit basis for
the commutant with respect to the Cartan subalgebra is obtained, and the order
of the polynomial algebra is computed. It is further shown that, with an
appropriate realization of $\mathfrak{sl}(n)$, this provides an explicit
connection with the generic superintegrable model on the $(n-1)$-dimensional
sphere $\mathbb{S}^{n-1}$ and the related Racah algebra $R(n)$. In particular,
we show explicitly how the models on the $2$-sphere and $3$-sphere and the
associated symmetry algebras can be obtained from the quadratic and cubic
polynomial algebras generated by the commutants defined in the enveloping
algebra of $\mathfrak{sl}(3)$ and $\mathfrak{sl}(4)$, respectively. The
construction is performed in the classical (or Poisson-Lie) context, where the
Berezin bracket replaces the commutator. | math-ph |
On the relativistic Vlasov-Poisson system: The Cauchy problem is revisited for the so-called relativistic Vlasov-Poisson
system in the attractive case. Global existence and uniqueness of spherical
classical solutions is proved under weaker assumptions than previously used. A
new class of blowing up solutions is found when these conditions are violated.
A new, non-gravitational physical vindication of the model which (unlike the
gravitational one) is not restricted to weak fields, is also given. | math-ph |
Topological Bragg Peaks And How They Characterise Point Sets: Bragg peaks in point set diffraction show up as eigenvalues of a dynamical
system. Topological Bragg peaks arrise from topological eigenvalues and
determine the torus parametrisation of the point set. We will discuss how
qualitative properties of the torus parametrisation characterise the point set. | math-ph |
Geometry and stability of dynamical systems: We reconsider both the global and local stability of solutions of
continuously evolving dynamical systems from a geometric perspective. We
clarify that an unambiguous definition of stability generally requires the
choice of additional geometric structure that is not intrinsic to the dynamical
system itself. While global Lyapunov stability is based on the choice of
seminorms on the vector bundle of perturbations, we propose a definition of
local stability based on the choice of a linear connection. We show how this
definition reproduces known stability criteria for second order dynamical
systems. In contrast to the general case, the special geometry of Lagrangian
systems provides completely intrinsic notions of global and local stability. We
demonstrate that these do not suffer from the limitations occurring in the
analysis of the Maupertuis-Jacobi geodesics associated to natural Lagrangian
systems. | math-ph |
Limiting distribution of extremal eigenvalues of d-dimensional random
Schrödinger operator: We consider Schr\"odinger operator with random decaying potential on $\ell^2
({\bf Z}^d)$ and showed that, (i) IDS coincides with that of free Laplacian in
general cases, and (ii) the set of extremal eigenvalues, after rescaling,
converges to a inhomogeneous Poisson process, under certain condition on the
single-site distribution, and (iii) there are "border-line" cases, such that we
have Poisson statistics in the sense of (ii) above if the potential does not
decay, while we do not if the potential does decay. | math-ph |
Classification of topological phases with finite internal symmetries in
all dimensions: We develop a mathematical theory of symmetry protected trivial (SPT) orders
and anomaly-free symmetry enriched topological (SET) orders in all dimensions
via two different approaches with an emphasis on the second approach. The first
approach is to gauge the symmetry in the same dimension by adding topological
excitations as it was done in the 2d case, in which the gauging process is
mathematically described by the minimal modular extensions of unitary braided
fusion 1-categories. This 2d result immediately generalizes to all dimensions
except in 1d, which is treated with special care. The second approach is to use
the 1-dimensional higher bulk of the SPT/SET order and the boundary-bulk
relation. This approach also leads us to a precise mathematical description and
a classification of SPT/SET orders in all dimensions. The equivalence of these
two approaches, together with known physical results, provides us with many
precise mathematical predictions. | math-ph |
How Lagrangian states evolve into random waves: In this paper, we consider a compact manifold $(X,d)$ of negative curvature,
and a family of semiclassical Lagrangian states $f_h(x) = a(x) e^{\frac{i}{h}
\phi(x)}$ on $X$. For a wide family of phases $\phi$, we show that $f_h$, when
evolved by the semiclassical Schr\"odinger equation during a long time,
resembles a random Gaussian field. This can be seen as an analogue of Berry's
random waves conjecture for Lagrangian states. | math-ph |
Gauge theories in noncommutative geometry: In this review we present some of the fundamental mathematical structures
which permit to define noncommutative gauge field theories. In particular, we
emphasize the theory of noncommutative connections, with the notions of
curvatures and gauge transformations. Two different approaches to
noncommutative geometry are covered: the one based on derivations and the one
based on spectral triples. Examples of noncommutative gauge field theories are
given to illustrate the constructions and to display some of the common
features. | math-ph |
Energy levels of neutral atoms via a new perturbation method: The energy levels of neutral atoms supported by Yukawa potential, $V(r)=-Z
exp(-\alpha r)/r$, are studied, using both dimensional and dimensionless
quantities, via a new analytical methodical proposal (devised to solve for
nonexactly solvable Schrodinger equation). Using dimensionless quantities, by
scaling the radial Hamiltonian through $y=Zr$ and $\alpha^{'}=\alpha/Z$, we
report that the scaled screening parameter $\alpha^{'}$ is restricted to have
values ranging from zero to less than 0.4. On the other hand, working with the
scaled Hamiltonian enhances the accuracy and extremely speeds up the
convergence of the energy eigenvalues. The energy levels of several new
eligible scaled screening parameter $\alpha^{'}$ values are also reported. | math-ph |
Langevin equations in the small-mass limit: Higher-order approximations: We study the small-mass (overdamped) limit of Langevin equations for a
particle in a potential and/or magnetic field with matrix-valued and
state-dependent drift and diffusion. We utilize a bootstrapping argument to
derive a hierarchy of approximate equations for the position degrees of freedom
that are able to achieve accuracy of order $m^{\ell/2}$ over compact time
intervals for any $\ell\in\mathbb{Z}^+$. This generalizes prior derivations of
the homogenized equation for the position degrees of freedom in the $m\to 0$
limit, which result in order $m^{1/2}$ approximations. Our results cover
bounded forces, for which we prove convergence in $L^p$ norms, and unbounded
forces, in which case we prove convergence in probability. | math-ph |
Alternative perturbation approaches in classical mechanics: We discuss two alternative methods, based on the Lindstedt--Poincar\'{e}
technique, for the removal of secular terms from the equations of perturbation
theory. We calculate the period of an anharmonic oscillator by means of both
approaches and show that one of them is more accurate for all values of the
coupling constant. | math-ph |
Fractional Dynamics of Systems with Long-Range Space Interaction and
Temporal Memory: Field equations with time and coordinates derivatives of noninteger order are
derived from stationary action principle for the cases of power-law memory
function and long-range interaction in systems. The method is applied to obtain
a fractional generalization of the Ginzburg-Landau and nonlinear Schrodinger
equations. As another example, dynamical equations for particles chain with
power-law interaction and memory are considered in the continuous limit. The
obtained fractional equations can be applied to complex media with/without
random parameters or processes. | math-ph |
Existence and measure of ergodic leaves in Novikov's problem on the
semiclassical motion of an electron: We show that ``ergodic regime'' appears for generic dispersion relations in
the semiclassical motion of electrons in a metal and we prove that, in the
fixed energy picture, the measure of the set of such directions is zero. | math-ph |
The sine process under the influence of a varying potential: We review the authors' recent work \cite{BDIK1,BDIK2,BDIK3} where we obtain
the uniform large $s$ asymptotics for the Fredholm determinant
$D(s,\gamma):=\det(I-\gamma K_s\upharpoonright_{L^2(-1,1)})$, $0\leq\gamma\leq
1$. The operator $K_s$ acts with kernel $K_s(x,y)=\sin(s(x-y))/(\pi(x-y))$ and
$D(s,\gamma)$ appears for instance in Dyson's model \cite{Dyson2} of a Coulomb
log-gas with varying external potential or in the bulk scaling analysis of the
thinned GUE \cite{BP}. | math-ph |
A Complete Basis for a Perturbation Expansion of the General N-Body
Problem: We discuss a basis set developed to calculate perturbation coefficients in an
expansion of the general N-body problem. This basis has two advantages. First,
the basis is complete order-by-order for the perturbation series. Second, the
number of independent basis tensors spanning the space for a given order does
not scale with N, the number of particles, despite the generality of the
problem. At first order, the number of basis tensors is 23 for all N although
the problem at first order scales as N^6. The perturbation series is expanded
in inverse powers of the spatial dimension. This results in a maximally
symmetric configuration at lowest order which has a point group isomorphic with
the symmetric group, S_N. The resulting perturbation series is order-by-order
invariant under the N! operations of the S_N point group which is responsible
for the slower than exponential growth of the basis. In this paper, we perform
the first test of this formalism including the completeness of the basis
through first order by comparing to an exactly solvable fully-interacting
problem of N particles with a two-body harmonic interaction potential. | math-ph |
On the space of light rays of a space-time and a reconstruction theorem
by Low: A reconstruction theorem in terms of the topology and geometrical structures
on the spaces of light rays and skies of a given space-time is discussed. This
result can be seen as part of Penrose and Low's programme intending to describe
the causal structure of a space-time $M$ in terms of the topological and
geometrical properties of the space of light rays, i.e., unparametrized
time-oriented null geodesics, $\mathcal{N}$. In the analysis of the
reconstruction problem it becomes instrumental the structure of the space of
skies, i.e., of congruences of light rays. It will be shown that the space of
skies $\Sigma$ of a strongly causal skies distinguishing space-time $M$ carries
a canonical differentiable structure diffeomorphic to the original manifold
$M$. Celestial curves, this is, curves in $\mathcal{N}$ which are everywhere
tangent to skies, play a fundamental role in the analysis of the geometry of
the space of light rays. It will be shown that a celestial curve is induced by
a past causal curve of events iff the legendrian isotopy defined by it is
non-negative. This result extends in a nontrivial way some recent results by
Chernov \emph{et al} on Low's Legendrian conjecture. Finally, it will be shown
that a celestial causal map between the space of light rays of two strongly
causal spaces (provided that the target space is null non-conjugate) is
necessarily induced from a conformal immersion and conversely. These results
make explicit the fundamental role played by the collection of skies, a
collection of legendrian spheres with respect to the canonical contact
structure on $\mathcal{N}$, in characterizing the causal structure of
space-times. | math-ph |
Local Central Limit Theorem for Determinantal Point Processes: We prove a local central limit theorem (LCLT) for the number of points $N(J)$
in a region $J$ in $\mathbb R^d$ specified by a determinantal point process
with an Hermitian kernel. The only assumption is that the variance of $N(J)$
tends to infinity as $|J| \to \infty$. This extends a previous result giving a
weaker central limit theorem (CLT) for these systems. Our result relies on the
fact that the Lee-Yang zeros of the generating function for $\{E(k;J)\}$ ---
the probabilities of there being exactly $k$ points in $J$ --- all lie on the
negative real $z$-axis. In particular, the result applies to the scaled bulk
eigenvalue distribution for the Gaussian Unitary Ensemble (GUE) and that of the
Ginibre ensemble. For the GUE we can also treat the properly scaled edge
eigenvalue distribution. Using identities between gap probabilities, the LCLT
can be extended to bulk eigenvalues of the Gaussian Symplectic Ensemble (GSE).
A LCLT is also established for the probability density function of the $k$-th
largest eigenvalue at the soft edge, and of the spacing between $k$-th neigbors
in the bulk. | math-ph |
A relativistic model of the $N$-dimensional singular oscillator: Exactly solvable $N$-dimensional model of the quantum isotropic singular
oscillator in the relativistic configurational $\vec r_N$-space is proposed. It
is shown that through the simple substitutions the finite-difference equation
for the $N$-dimensional singular oscillator can be reduced to the similar
finite-difference equation for the relativistic isotropic three-dimensional
singular oscillator. We have found the radial wavefunctions and energy spectrum
of the problem and constructed a dynamical symmetry algebra. | math-ph |
A numerical approach to harmonic non-commutative spectral field theory: We present a first numerical investigation of a non-commutative gauge theory
defined via the spectral action for Moyal space with harmonic propagation. This
action is approximated by finite matrices. Using Monte Carlo simulation we
study various quantities such as the energy density, the specific heat density
and some order parameters, varying the matrix size and the independent
parameters of the model. We find a peak structure in the specific heat which
might indicate possible phase transitions. However, there are mathematical
arguments which show that the limit of infinite matrices is very different from
the original spectral model. | math-ph |
Exact Fractional Revival in Spin Chains: The occurrence of fractional revival in quantum spin chains is examined.
Analytic models where this phenomenon can be exhibited in exact solutions are
provided. It is explained that spin chains with fractional revival can be
obtained by isospectral deformations of spin chains with perfect state
transfer. | math-ph |
Quantization, Dequantization, and Distinguished States: Geometric quantization is a natural way to construct quantum models starting
from classical data. In this work, we start from a symplectic vector space with
an inner product and -- using techniques of geometric quantization -- construct
the quantum algebra and equip it with a distinguished state. We compare our
result with the construction due to Sorkin -- which starts from the same input
data -- and show that our distinguished state coincides with the Sorkin-Johnson
state. Sorkin's construction was originally applied to the free scalar field
over a causal set (locally finite, partially ordered set). Our perspective
suggests a natural generalization to less linear examples, such as an
interacting field. | math-ph |
Parametrizations of degenerate density matrices: It turns out that a parametrization of degenerate density matrices requires a
parametrization of $\mathfrak{F}=U(n)/({U(k_1)\times U(k_2)\times \cdots \times
U(k_m)})\quad n=k_1 +\cdots + k_m $ where $U(k)$ denotes the set of all unitary
$k\times k$-matrices with complex entries. Unfortunately the parametrization of
this quotient space is quite involved. Our solution does not rely on Lie
algebra methods {directly,} but succeeds through the construction of suitable
sections for natural projections, by using techniques from the theory of
homogeneous spaces. We mention the relation to the Lie algebra back ground and
conclude with two concrete examples. | math-ph |
Self-adjointness and domain of generalized spin-boson models with mild
ultraviolet divergences: We provide a rigorous construction of a large class of generalized spin-boson
models with ultraviolet-divergent form factors. This class comprises various
models of many possibly non-identical atoms with arbitrary but finite numbers
of levels, interacting with a boson field. Ultraviolet divergences are assumed
to be mild, such that no self-energy renormalization is necessary. Our
construction is based on recent results by A. Posilicano, which also allow us
to state an explicit formula for the domain of self-adjointness for our
Hamiltonians. | math-ph |
Exact solutions with singularities to ideal hydrodynamics of inelastic
gases: We construct a large family of exact solutions to the hyperbolic system of 3
equations of ideal granular hydrodynamics in several dimensions for arbitrary
adiabatic index $\gamma$. In dependence of initial conditions these solutions
can keep smoothness for all times or develop singularity. In particular, in the
2D case the singularity can be formed either in a point or along a line. For
$\gamma=-1$ the problem is reduced to the system of two equations, related to a
special case of the Chaplygin gas. In the 1D case this system can be written in
the Riemann invariant and can be treated in a standard way. The solution to the
Riemann problem in this case demonstrate an unusual and complicated behavior. | math-ph |
Bocher contractions of conformally superintegrable Laplace equations:
Detailed computations: These supplementary notes in the ArXiv are a companion to our paper "Bocher
contractions of conformally superintegrable Laplace equations"
[arXiv:1512.09315]. They contain background material and the details of the
extensive computations that couldn't be put in the paper, due to space
limitations. | math-ph |
Seven-body central configurations: a family of central configurations in
the spatial seven-body problem: The main result of this paper is the existence of a new family of central
configurations in the Newtonian spatial seven-body problem. This family is
unusual in that it is a simplex stacked central configuration, i.e the bodies
are arranged as concentric three and two dimensional simplexes. | math-ph |
Analytical Evaluation Of An Infinite Integral Over Four Spherical Bessel
Functions: An infinite integral over four spherical Bessel functions is analytically
evaluated for the special case when the arguments k_3=k_1 and k_4=k_2 | math-ph |
Design of high-order short-time approximations as a problem of matching
the covariance of a Brownian motion: One of the outstanding problems in the numerical discretization of the
Feynman-Kac formula calls for the design of arbitrary-order short-time
approximations that are constructed in a stable way, yet only require knowledge
of the potential function. In essence, the problem asks for the development of
a functional analogue to the Gauss quadrature technique for one-dimensional
functions. In PRE 69, 056701 (2004), it has been argued that the problem of
designing an approximation of order \nu is equivalent to the problem of
constructing discrete-time Gaussian processes that are supported on
finite-dimensional probability spaces and match certain generalized moments of
the Brownian motion. Since Gaussian processes are uniquely determined by their
covariance matrix, it is tempting to reformulate the moment-matching problem in
terms of the covariance matrix alone. Here, we show how this can be
accomplished. | math-ph |
On one photon scattering in non-relativistic qed: We consider scattering of a single photon by an atom or a molecule in the
framework of non relativistic qed, and we express the scattering matrix for one
photon scattering as a boundary value of the resolvent. | math-ph |
Tagged particle process in continuum with singular interactions: By using Dirichlet form techniques we construct the dynamics of a tagged
particle in an infinite particle environment of interacting particles for a
large class of interaction potentials. In particular, we can treat interaction
potentials having a singularity at the origin, non-trivial negative part and
infinite range, as e.g., the Lennard-Jones potential. | math-ph |
Operator reflection positivity inequalities and their applications to
interacting quantum rotors: In the Reflection Positivity theory and its application to statistical
mechanical systems, certain matrix inequalities play a central role. The
Dyson-Lieb-Simon and Kennedy-Lieb-Shastry-Schupp inequalities constitute
prominent examples. In this paper we extend the KLS-S inequality to the case
where matrices are replaced by certain operators. As an application, we prove
the occurrence of the long range order in the ground state of two-dimensional
quantum rotors. | math-ph |
Hypergeometric First Integrals of the Duffing and van der Pol
Oscillators: The autonomous Duffing oscillator, and its van der Pol modification, are
known to admit time-dependent first integrals for specific values of
parameters. This corresponds to the existence of Darboux polynomials, and in
fact more can be shown: that there exist Liouvillian first integrals which do
not depend on time. They can be expressed in terms of the Gauss and Kummer
hypergeometric functions, and are neither analytic, algebraic nor meromorphic.
A criterion for this to happen in a general dynamical system is formulated as
well. | math-ph |
Weak singularity dynamics in a nonlinear viscous medium: We consider a system of nonlinear equations which can be reduced to a
degenerate parabolic equation. In the case $x\in\bR^2$ we obtained necessary
conditions for the existence of a weakly singular solution of heat wave type
($\codim\sing\supp=1$) and of vortex type ($\codim\sing\supp=2$). These
conditions have the form of a sequence of differential equations and allow one
to calculate the dynamics of the singularity support. In contrast to the
methods used traditionally for degenerate parabolic equations, our approach is
not based on comparison theorems. | math-ph |
Comment on "Design of acoustic devices with isotropic material via
conformal transformation" [Appl. Phys. Lett. 97, 044101 (2010)]: The paper presents incorrect formulas for the density and bulk modulus under
a conformal transformation of coordinates. The fault lies with an improper
assumption of constant acoustic impedance. | math-ph |
Global gauge conditions in the Batalin-Vilkovisky formalism: In the Batalin-Vilkovisky formalism, gauge conditions are expressed as
Lagrangian submanifolds in the space of fields and antifields. We discuss a way
of patching together gauge conditions over different parts of the space of
fields, and apply this method to extend the light-cone gauge for the
superparticle to a conic neighbourhood of the forward light-cone in momentum
space. | math-ph |
Dynamical system induced by quantum walk: We consider the Grover walk model on a connected finite graph with two
infinite length tails and we set an $\ell^\infty$-infinite external source from
one of the tails as the initial state. We show that for any connected internal
graph, a stationary state exists, moreover a perfect transmission to the
opposite tail always occurs in the long time limit. We also show that the lower
bound of the norm of the stationary measure restricted to the internal graph is
proportion to the number of edges of this graph. Furthermore when we add more
tails (e.g., $r$-tails) to the internal graph, then we find that from the
temporal and spatial global view point, the scattering to each tail in the long
time limit coincides with the local one-step scattering manner of the Grover
walk at a vertex whose degree is $(r+1)$. | math-ph |
On the Ground State Energy of the Delta-Function Fermi Gas II: Further
Asymptotics: Building on previous work of the authors, we here derive the weak coupling
asymptotics to order $\gamma^2$ of the ground state energy of the
delta-function Fermi gas. We use a method that can be applied to a large class
of finite convolution operators. | math-ph |
A Contour Integral Representation for the Dual Five-Point Function and a
Symmetry of the Genus Four Surface in R6: The invention of the "dual resonance model" N-point functions BN motivated
the development of current string theory. The simplest of these models, the
four-point function B4, is the classical Euler Beta function. Many standard
methods of complex analysis in a single variable have been applied to elucidate
the properties of the Euler Beta function, leading, for example, to analytic
continuation formulas such as the contour-integral representation obtained by
Pochhammer in 1890. Here we explore the geometry underlying the dual five-point
function B5, the simplest generalization of the Euler Beta function. Analyzing
the B5 integrand leads to a polyhedral structure for the five-crosscap surface,
embedded in RP5, that has 12 pentagonal faces and a symmetry group of order 120
in PGL(6). We find a Pochhammer-like representation for B5 that is a contour
integral along a surface of genus five. The symmetric embedding of the
five-crosscap surface in RP5 is doubly covered by a symmetric embedding of the
surface of genus four in R6 that has a polyhedral structure with 24 pentagonal
faces and a symmetry group of order 240 in O(6). The methods appear
generalizable to all N, and the resulting structures seem to be related to
associahedra in arbitrary dimensions. | math-ph |
Phase Transitions in Long-Range Random Field Ising Models in Higher
Dimensions: We extend the recent argument by Ding and Zhuang from nearest-neighbor to
long-range interactions and prove the phase transition in the class of
ferromagnetic random field Ising models. Our proof combines a generalization of
Fr\"ohlich-Spencer contours to the multidimensional setting, proposed by two of
us, with the coarse-graining procedure introduced by Fisher, Fr\"ohlich and
Spencer. The result shows that the Ding-Zhuang strategy is also useful for
interactions $J_{xy}=|x-y|^{- \alpha}$ when $\alpha > d$ in dimension $d\geq 3$
if we have a suitable system of contours. We can consider i.i.d. random fields
with Gaussian or Bernoulli distributions. Our main result is an alternative
proof that does not use the Renormalization Group Method (RGM), since Bricmont
and Kupiainen claimed that the RGM should also work on this generality. | math-ph |
Translation-invariant and periodic Gibbs measures for Potts model on a
Cayley tree: In this paper is studied ferromagnetic three states Potts model on a Cayley
tree of order three and we give explicit formulas for translation-invariant
Gibbs measures. Furthermore, we show that under some conditions on the
parameter of the antiferromagnetic Potts model with q-states with zero external
field on the Cayley tree of order $k>2$, there are exactly 2(2^q-1) periodic
(non translation-invariant) Gibbs measures. | math-ph |
Describing certain Lie algebra orbits via polynomial equations: Let $\mathfrak{h}_3$ be the Heisenberg algebra and let $\mathfrak g$ be the
3-dimensional Lie algebra having $[e_1,e_2]=e_1\,(=-[e_2,e_1])$ as its only
non-zero commutation relations. We describe the closure of the orbit of a
vector of structure constants corresponding to $\mathfrak{h}_3$ and $\mathfrak
g$ respectively as an algebraic set giving in each case a set of polynomials
for which the orbit closure is the set of common zeros. Working over an
arbitrary infinite field, this description enables us to give an alternative
way, using the definition of an irreducible algebraic set, of obtaining all
degenerations of $\mathfrak{h}_3$ and $\mathfrak g$ (the degeneration from
$\mathfrak g$ to $\mathfrak{h}_3$ being one of them). | math-ph |
A new generalisation of Macdonald polynomials: We introduce a new family of symmetric multivariate polynomials, whose
coefficients are meromorphic functions of two parameters $(q,t)$ and polynomial
in a further two parameters $(u,v)$. We evaluate these polynomials explicitly
as a matrix product. At $u=v=0$ they reduce to Macdonald polynomials, while at
$q=0$, $u=v=s$ they recover a family of inhomogeneous symmetric functions
originally introduced by Borodin. | math-ph |
Wave kernel for the Schrodinger operator with a Liouville potential: In this note we give an explicit formula for the wave equation associated to
the Schrodinger operator with a Liouville Potential with applications to the
telegraph equation as well as the wave equation on the hyperbolic plane | math-ph |
Discrete Energy Asymptotics on a Riemannian circle: We derive the complete asymptotic expansion in terms of powers of $N$ for the
geodesic $f$-energy of $N$ equally spaced points on a rectifiable simple closed
curve $\Gamma$ in ${\mathbb R}^p$, $p\geq2$, as $N \to \infty$. For $f$
decreasing and convex, such a point configuration minimizes the $f$-energy
$\sum_{j\neq k}f(d(\mathbf{x}_j, \mathbf{x}_k))$, where $d$ is the geodesic
distance (with respect to $\Gamma$) between points on $\Gamma$. Completely
monotonic functions, analytic kernel functions, Laurent series, and weighted
kernel functions $f$ are studied. % Of particular interest are the geodesic
Riesz potential $1/d^s$ ($s \neq 0$) and the geodesic logarithmic potential
$\log(1/d)$. By analytic continuation we deduce the expansion for all complex
values of $s$. | math-ph |
Vortex pairs and dipoles on closed surfaces: We set up general equations of motion for point vortex systems on closed
Riemannian surfaces, allowing for the case that the sum of vorticities is not
zero and there hence must be counter-vorticity present. The dynamics of global
circulations which is coupled to the dynamics of the vortices is carefully
taken into account.
Much emphasis is put to the study of vortex pairs, having the Kimura
conjecture in focus. This says that vortex pairs move, in the dipole limit,
along geodesic curves, and proofs for it have previously been given by
S.~Boatto and J.~Koiller by using Gaussian geodesic coordinates. In the present
paper we reach the same conclusion by following a slightly different route,
leading directly to the geodesic equation with a reparametrized time variable.
In a final section we explain how vortex motion in planar domains can be seen
as a special case of vortex motion on closed surfaces, and in two appendices we
give some necessary background on affine and projective connections. | math-ph |
Lie subalgebras of the matrix quantum pseudo differential operators: We give a complete description of the anti-involutions that preserve the
principal gradation of the algebra of matrix quantum pseudodifferential
operators and we describe the Lie subalgebras of its minus fixed points. | math-ph |
Post-Processing Enhancement of Reverberation-Noise Suppression in
Dual-Frequency SURF Imaging: A post-processing adjustment technique which aims for enhancement of
dual-frequency SURF (Second order UltRasound Field) reverberation-noise
suppression imaging in medical ultrasound is analyzed. Two variant methods are
investigated through numerical simulations. They both solely involve
post-processing of the propagated high-frequency (HF) imaging wave fields,
which in real-time imaging corresponds to post-processing of the beamformed
receive radio-frequency signals. Hence the transmit pulse complexes are the
same as for the previously published SURF reverberation-suppression imaging
method. The adjustment technique is tested on simulated data from propagation
of SURF pulse complexes consisting of a 3.5 MHz HF imaging pulse added to a 0.5
low-frequency sound-speed manipulation pulse. Imaging transmit beams are
constructed with and without adjustment. The post-processing involves
filtering, e.g., by a time-shift, in order to equalize the two SURF HF pulses
at a chosen depth. This depth is typically chosen to coincide with the depth
where the first scattering or reflection occurs for the reverberation noise one
intends to suppress. The beams realized with post-processing show energy
decrease at the chosen depth, especially for shallow depths where in a medical
imaging situation often a body-wall is located. This indicates that the
post-processing may further enhance the reverberation-suppression abilities of
SURF imaging. Moreover, it is shown that the methods might be utilized to
reduce the accumulated near-field energy of the SURF transmit-beam relative to
its imaging region energy. The adjustments presented may therefore potentially
be utilized to attain a slightly better general suppression of multiple
scattering and multiple reflection noise compared to for non-adjusted SURF
reverberation-suppression imaging. | math-ph |
Affine geometric description of thermodynamics: Thermodynamics provides a unified perspective of thermodynamic properties of
various substances. To formulate thermodynamics in the language of
sophisticated mathematics, thermodynamics is described by a variety of
differential geometries, including contact and symplectic geometries. Meanwhile
affine geometry is a branch of differential geometry and is compatible with
information geometry, where information geometry is known to be compatible with
thermodynamics. By combining above, it is expected that thermodynamics is
compatible with affine geometry, and is expected that several affine geometric
tools can be introduced in the analysis of thermodynamic systems. In this paper
affine geometric descriptions of equilibrium and nonequilibrium thermodynamics
are proposed. For equilibrium systems, it is shown that several thermodynamic
quantities can be identified with geometric objects in affine geometry, and
that several geometric objects can be introduced in thermodynamics. Examples of
these include: specific heat is identified with the affine fundamental form, a
flat connection is introduced in thermodynamic phase space. For nonequilibrium
systems, two classes of relaxation processes are shown to be described in the
language of an extension of affine geometry. Finally this affine geometric
description of thermodynamics for equilibrium and nonequilibrium systems is
compared with a contact geometric description. | math-ph |
Wu-Yang ambiguity in connection space: Two distinct gauge potentials can have the same field strength, in which case
they are said to be ``copies'' of each other. The consequences of this
possibility for the general space A of gauge potentials are examined. Any two
potentials are connected by a straight line in A, but a straight line going
through two copies either contains no other copy or is entirely formed by
copies. | math-ph |
Thermodynamic limit and twisted boundary energy of the XXZ spin chain
with antiperiodic boundary condition: We investigate the thermodynamic limit of the inhomogeneous T-Q relation of
the antiferromagnetic XXZ spin chain with antiperiodic boundary condition. It
is shown that the contribution of the inhomogeneous term at the ground state
can be neglected when the system-size N tends to infinity, which enables us to
reduce the inhomogeneous Bethe ansatz equations (BAEs) to the homogeneous ones.
Then the quantum numbers at the ground states are obtained, by which the system
with arbitrary size can be studied. We also calculate the twisted boundary
energy of the system. | math-ph |
Recursion for the smallest eigenvalue density of
$β$-Wishart-Laguerre ensemble: The statistics of the smallest eigenvalue of Wishart-Laguerre ensemble is
important from several perspectives. The smallest eigenvalue density is
typically expressible in terms of determinants or Pfaffians. These results are
of utmost significance in understanding the spectral behavior of
Wishart-Laguerre ensembles and, among other things, unveil the underlying
universality aspects in the asymptotic limits. However, obtaining exact and
explicit expressions by expanding determinants or Pfaffians becomes impractical
if large dimension matrices are involved. For the real matrices ($\beta=1$)
Edelman has provided an efficient recurrence scheme to work out exact and
explicit results for the smallest eigenvalue density which does not involve
determinants or matrices. Very recently, an analogous recurrence scheme has
been obtained for the complex matrices ($\beta=2$). In the present work we
extend this to $\beta$-Wishart-Laguerre ensembles for the case when exponent
$\alpha$ in the associated Laguerre weight function, $\lambda^\alpha
e^{-\beta\lambda/2}$, is a non-negative integer, while $\beta$ is positive
real. This also gives access to the smallest eigenvalue density of fixed trace
$\beta$-Wishart-Laguerre ensemble, as well as moments for both cases. Moreover,
comparison with earlier results for the smallest eigenvalue density in terms of
certain hypergeometric function of matrix argument results in an effective way
of evaluating these explicitly. Exact evaluations for large values of $n$ (the
matrix dimension) and $\alpha$ also enable us to compare with Tracy-Widom
density and large deviation results of Katzav and Castillo. We also use our
result to obtain the density of the largest of the proper delay times which are
eigenvalues of the Wigner-Smith matrix and are relevant to the problem of
quantum chaotic scattering. | math-ph |
The inverse Rytov series for diffuse optical tomography: The Rytov approximation is known in near-infrared spectroscopy including
diffuse optical tomography. In diffuse optical tomography, the Rytov
approximation often gives better reconstructed images than the Born
approximation. Although related inverse problems are nonlinear, the Rytov
approximation is almost always accompanied by the linearization of nonlinear
inverse problems. In this paper, we will develop nonlinear reconstruction with
the inverse Rytov series. By this, linearization is not necessary and higher
order terms in the Rytov series can be used for reconstruction. The convergence
and stability are discussed. We find that the inverse Rytov series has a
recursive structure similar to the inverse Born series. | math-ph |
Beta Deformation and Superpolynomials of (n,m) Torus Knots: Recent studies in several interrelated areas -- from combinatorics and
representation theory in mathematics to quantum field theory and topological
string theory in physics -- have independently revealed that many classical
objects in these fields admit a relatively novel one-parameter deformation.
This deformation, known in different contexts under the names of
Omega-background, refinement, or beta-deformation, has a number of interesting
mathematical implications. In particular, in Chern-Simons theory
beta-deformation transforms the classical HOMFLY invariants into
Dunfield-Gukov-Rasmussen superpolynomials -- Poincare polynomials of a triply
graded knot homology theory. As shown in arXiv:1106.4305, these
superpolynomials are particular linear combinations of rational Macdonald
dimensions, distinguished by the polynomiality, integrality and positivity
properties. We show that these properties alone do not fix the superpolynomials
uniquely, by giving an example of a combination of Macdonald dimensions, that
is always a positive integer polynomial but generally is not a superpolynomial. | math-ph |
Noether conservation laws in classical mechanics: In Lagrangian mechanics, Noether conservation laws including the energy one
are obtained similarly to those in field theory. In Hamiltonian mechanics,
Noether conservation laws are issued from the invariance of the Poincare-Cartan
integral invariant under one-parameter groups of diffeomorphisms of a
configuration space. Lagrangian and Hamiltonian conservation laws need not be
equivalent. | math-ph |
A summation formula over the zeros of a combination of the associated
Legendre functions with a physical application: By using the generalized Abel-Plana formula, we derive a summation formula
for the series over the zeros of a combination of the associated Legendre
functions with respect to the degree. The summation formula for the series over
the zeros of the combination of the Bessel functions, previously discussed in
the literature, is obtained as a limiting case. As an application we evaluate
the Wightman function for a scalar field with general curvature coupling
parameter in the region between concentric spherical shells on background of
constant negative curvature space. For the Dirichlet boundary conditions the
corresponding mode-sum contains series over the zeros of the combination of the
associated Legendre functions. The application of the summation formula allows
us to present the Wightman function in the form of the sum of two integrals.
The first one corresponds to the Wightman function for the geometry of a single
spherical shell and the second one is induced by the presence of the second
shell. The boundary-induced part in the vacuum expectation value of the field
squared is investigated. For points away from the boundaries the corresponding
renormalization procedure is reduced to that for the boundary-free part. | math-ph |
Note on the Relativistic Thermodynamics of Moving Bodies: We employ a novel thermodynamical argument to show that, at the macroscopic
level,there is no intrinsic law of temperature transformation under Lorentz
boosts. This result extends the corresponding microstatistical one of earlier
works to the purely macroscopic regime and signifies that the concept of
temperature as an objective entity is restricted to the description of bodies
in their rest frames. The argument on which this result is based is centred on
the thermal transactions between a body that moves with uniform velocity
relative to a certain inertial frame and a thermometer, designed to measure its
temperature, that is held at rest in that frame. | math-ph |
Extensions of diffeomorphism and current algebras: Dzhumadil'daev has classified all tensor module extensions of $diff(N)$, the
diffeomorphism algebra in $N$ dimensions, and its subalgebras of divergence
free, Hamiltonian, and contact vector fields. I review his results using
explicit tensor notation. All of his generic cocycles are limits of trivial
cocycles, and many arise from the Mickelsson-Faddeev algebra for $gl(N)$. Then
his results are extended to some non-tensor modules, including the
higher-dimensional Virasoro algebras found by Eswara Rao/Moody and myself.
Extensions of current algebras with $d$-dimensional representations are
obtained by restriction from $diff(N+d)$. This gives a connection between
higher-dimensional Virasoro and Kac-Moody cocycles, and between
Mickelsson-Faddeev cocycles for diffeomorphism and current algebras. | math-ph |
Solving the Navier-Lame Equation in Cylindrical Coordinates Using the
Buchwald Representation: Some Parametric Solutions with Applications: Using a separable Buchwald representation in cylindrical coordinates, we show
how under certain conditions the coupled equations of motion governing the
Buchwald potentials can be decoupled and then solved using well-known
techniques from the theory of PDEs. Under these conditions, we then construct
three parametrized families of particular solutions to the Navier-Lame equation
in cylindrical coordinates. In this paper, we specifically construct solutions
having 2pi-periodic angular parts. These particular solutions can be directly
applied to a fundamental set of linear elastic boundary value problems in
cylindrical coordinates and are especially suited to problems involving one or
more physical parameters. As an illustrative example, we consider the problem
of determining the response of a solid elastic cylinder subjected to a
time-harmonic surface pressure that varies sinusoidally along its axis, and we
demonstrate how the obtained parametric solutions can be used to efficiently
construct an exact solution to this problem. We also briefly consider
applications to some related forced-relaxation type problems. | math-ph |
Band gap of the Schroedinger operator with a strong delta-interaction on
a periodic curve: In this paper we study the operator
$H_{\beta}=-\Delta-\beta\delta(\cdot-\Gamma)$ in $L^{2}(\mathbb{R}^{2})$, where
$\Gamma$ is a smooth periodic curve in $\mathbb{R}^{2}$. We obtain the
asymptotic form of the band spectrum of $H_{\beta}$ as $\beta$ tends to
infinity. Furthermore, we prove the existence of the band gap of
$\sigma(H_{\beta})$ for sufficiently large $\beta>0$. Finally, we also derive
the spectral behaviour for $\beta\to\infty$ in the case when $\Gamma$ is
non-periodic and asymptotically straight. | math-ph |
An Obstruction to Quantization of the Sphere: In the standard example of strict deformation quantization of the symplectic
sphere $S^2$, the set of allowed values of the quantization parameter $\hbar$
is not connected; indeed, it is almost discrete. Li recently constructed a
class of examples (including $S^2$) in which $\hbar$ can take any value in an
interval, but these examples are badly behaved. Here, I identify a natural
additional axiom for strict deformation quantization and prove that it implies
that the parameter set for quantizing $S^2$ is never connected. | math-ph |
Some Results on Inverse Scattering: A review of some of the author's results in the area of inverse scattering is
given. The following topics are discussed: 1) Property $C$ and applications, 2)
Stable inversion of fixed-energy 3D scattering data and its error estimate, 3)
Inverse scattering with ''incomplete`` data, 4) Inverse scattering for
inhomogeneous Schr\"odinger equation, 5) Krein's inverse scattering method, 6)
Invertibility of the steps in Gel'fand-Levitan, Marchenko, and Krein inversion
methods, 7) The Newton-Sabatier and Cox-Thompson procedures are not inversion
methods, 8) Resonances: existence, location, perturbation theory, 9) Born
inversion as an ill-posed problem, 10) Inverse obstacle scattering with
fixed-frequency data, 11) Inverse scattering with data at a fixed energy and a
fixed incident direction, 12) Creating materials with a desired refraction
coefficient and wave-focusing properties. | math-ph |
Systems of coupled PT-symmetric oscillators: The Hamiltonian for a PT-symmetric chain of coupled oscillators is
constructed. It is shown that if the loss-gain parameter $\gamma$ is uniform
for all oscillators, then as the number of oscillators increases, the region of
unbroken PT-symmetry disappears entirely. However, if $\gamma$ is localized in
the sense that it decreases for more distant oscillators, then the
unbroken-PT-symmetric region persists even as the number of oscillators
approaches infinity. In the continuum limit the oscillator system is described
by a PT-symmetric pair of wave equations, and a localized loss-gain impurity
leads to a pseudo-bound state. It is also shown that a planar configuration of
coupled oscillators can have multiple disconnected regions of unbroken PT
symmetry. | math-ph |
Balance between quantum Markov semigroups: The concept of balance between two state preserving quantum Markov semigroups
on von Neumann algebras is introduced and studied as an extension of conditions
appearing in the theory of quantum detailed balance. This is partly motivated
by the theory of joinings. Balance is defined in terms of certain correlated
states (couplings), with entangled states as a specific case. Basic properties
of balance are derived and the connection to correspondences in the sense of
Connes is discussed. Some applications and possible applications, including to
non-equilibrium statistical mechanics, are briefly explored. | math-ph |
On Graph-Theoretic Identifications of Adinkras, Supersymmetry
Representations and Superfields: In this paper we discuss off-shell representations of N-extended
supersymmetry in one dimension, ie, N-extended supersymmetric quantum
mechanics, and following earlier work on the subject codify them in terms of
certain graphs, called Adinkras. This framework provides a method of generating
all Adinkras with the same topology, and so also all the corresponding
irreducible supersymmetric multiplets. We develop some graph theoretic
techniques to understand these diagrams in terms of a relatively small amount
of information, namely, at what heights various vertices of the graph should be
"hung".
We then show how Adinkras that are the graphs of N-dimensional cubes can be
obtained as the Adinkra for superfields satisfying constraints that involve
superderivatives. This dramatically widens the range of supermultiplets that
can be described using the superspace formalism and organizes them. Other
topologies for Adinkras are possible, and we show that it is reasonable that
these are also the result of constraining superfields using superderivatives.
The family of Adinkras with an N-cubical topology, and so also the sequence
of corresponding irreducible supersymmetric multiplets, are arranged in a
cyclical sequence called the main sequence. We produce the N=1 and N=2 main
sequences in detail, and indicate some aspects of the situation for higher N. | math-ph |
Semiclassical States on Lie Algebras: The effective technique for analyzing representation-independent features of
quantum systems based on the semiclassical approximation (developed elsewhere),
has been successfully used in the context of the canonical (Weyl) algebra of
the basic quantum observables. Here we perform the important step of extending
this effective technique to the quantization of a more general class of
finite-dimensional Lie algebras. The case of a Lie algebra with a single
central element (the Casimir element) is treated in detail by considering
semiclassical states on the corresponding universal enveloping algebra.
Restriction to an irreducible representation is performed by "effectively"
fixing the Casimir condition, following the methods previously used for
constrained quantum systems. We explicitly determine the conditions under which
this restriction can be consistently performed alongside the semiclassical
truncation. | math-ph |
Dynamical rigidity of stochastic Coulomb systems in infinite-dimensions: This paper is based on the talk in "Probability Symposium" at Research
Institute of Mathematical Sciences (Kyoto University) on 2013/12/18, and gives
an announcement of some parts of the results in [1,8,10,11]. We show two
instances of dynamical rigidity of Ginibre interacting Brownian motion in
infinite dimensions. This stochastic dynamics is given by the
infinite-dimensional stochastic differential equation describing infinite-many
Brownian particles in the plane interacting through two-dimensional Coulomb
potential. The first dynamical rigidity is that the Ginibre interacting
Brownian motion is a unique, strong solution of two different infinite
dimensional stochastic differential equations. The second shows that the tagged
particles of Ginibre interacting Brownian motion are sub diffusive. We also
propose the notion of "Coulomb random point fields" and the associated "Coulomb
interacting Brownian motions". | math-ph |
Scale and Möbius covariance in two-dimensional Haag-Kastler net: Given a two-dimensional Haag-Kastler net which is Poincar\'e-dilation
covariant with additional properties, we prove that it can be extended to a
M\"obius covariant net. Additional properties are either a certain condition on
modular covariance, or a variant of strong additivity. The proof relies neither
on the existence of stress-energy tensor nor any assumption on scaling
dimensions. We exhibit some examples of Poincar\'e-dilation covariant net which
cannot be extended to a M\"obius covariant net, and discuss the obstructions. | math-ph |
Relative equilibria and relative periodic solutions in systems with
time-delay and $S^{1}$ symmetry: We study properties of basic solutions in systems with dime delays and
$S^1$-symmetry. Such basic solutions are relative equilibria (CW solutions) and
relative periodic solutions (MW solutions). It follows from the previous theory
that the number of CW solutions grows generically linearly with time delay
$\tau$. Here we show, in particular, that the number of relative periodic
solutions grows generically as $\tau^2$ when delay increases. Thus, in such
systems, the relative periodic solutions are more abundant than relative
equilibria. The results are directly applicable to, e.g., Lang-Kobayashi model
for the lasers with delayed feedback. We also study stability properties of the
solutions for large delays. | math-ph |
On the Poincaré's generating function and the symplectic mid-point
rule: The use of Liouvillian forms to obtain symplectic maps for constructing
numerical integrators is a natural alternative to the method of generating
functions, and provides a deeper understanding of the geometry of this
procedure. Using Liouvillian forms we study the generating function introduced
by Poincar\'e (1899) and its associated symplectic map. We show that in this
framework, Poincar\'e's generating function does not correspond to the
symplectic mid-point rule, but to the identity map. We give an interpretation
of this result based on the original framework constructed by Poincar\'e. | math-ph |
A physics pathway to the Riemann hypothesis: We present a brief review of the spectral approach to the Riemann hypothesis,
according to which the imaginary part of the non trivial zeros of the zeta
function are the eigenvalues of the Hamiltonian of a quantum mechanical system. | math-ph |
Completely positive invariant conjugate-bilinear maps in partial
*-algebras: The notion of completely positive invariant conjugate-bilinear map in a
partial *-algebra is introduced and a generalized Stinespring theorem is
proven. Applications to the existence of integrable extensions of
*-representations of commutative, locally convex quasi*-algebras are also
discussed. | math-ph |
On anomalies in classical dynamical systems: The definition of "classical anomaly" is introduced. It describes the
situation in which a purely classical dynamical system which presents both a
lagrangian and a hamiltonian formulation admits symmetries of the action for
which the Noether conserved charges, endorsed with the Poisson bracket
structure, close an algebra which is just the centrally extended version of the
original symmetry algebra. The consistency conditions for this to occur are
derived. Explicit examples are given based on simple two-dimensional models.
Applications of the above scheme and lines of further investigations are
suggested. | math-ph |
Deficiency indices for singular magnetic Schrödinger operators: We show that the deficiency indices of magnetic Schr\"odinger operators with
several local singularities can be computed in terms of the deficiency indices
of operators carrying just one singularity each. We discuss some applications
to physically relevant operators. | math-ph |
Symplectic Non-Squeezing Theorems, Quantization of Integrable Systems,
and Quantum Uncertainty: The ground energy level of an oscillator cannot be zero because of
Heisenberg's uncertainty principle. We use methods from symplectic topology
(Gromov's non-squeezing theorem, and the existence of symplectic capacities) to
analyze and extend this heuristic observation to Liouville-integrable systems,
and to propose a topological quantization scheme for such systems, thus
extending previous results of ours. | math-ph |
Entropic Fluctuations in Quantum Statistical Mechanics. An Introduction: These lecture notes provide an elementary introduction, within the framework
of finite quantum systems, to recent developments in the theory of entropic
fluctuations. | math-ph |
The Painlevé analysis for N=2 super KdV equations: The Painlev\'e analysis of a generic multiparameter N=2 extension of the
Korteweg-de Vries equation is presented. Unusual aspects of the analysis,
pertaining to the presence of two fermionic fields, are emphasized. For the
general class of models considered, we find that the only ones which manifestly
pass the test are precisely the four known integrable supersymmetric KdV
equations, including the SKdV$_1$ case. | math-ph |
Diffusion Scattering of Waves is a Model of Subquantum Level?: In the paper, we discuss the studies of mathematical models of diffusion
scattering of waves in the phase space, and relation of these models with
quantum mechanics. In the previous works it is shown that in these models of
classical scattering process of waves, the quantum mechanical description
arises as the asymptotics after a small time. In this respect, the proposed
models can be considered as examples in which the quantum descriptions arise as
approximate ones for certain hypothetical reality. The deviation between the
proposed models and the quantum ones can arise, for example, for processes with
rapidly changing potential function. Under its action the diffusion scattering
process of waves will go out from the states described by quantum mechanics.
In the paper it is shown that the proposed models of diffusion scattering of
waves possess the property of gauge invariance. This implies that they are
described similarly in all inertial coordinate systems, i.e., they are
invariant under the Galileo transformations.
We propose a program of further research. | math-ph |
Feynman integrals as Hida distributions: the case of non-perturbative
potentials: Feynman integrands are constructed as Hida distributions. For our approach we
first have to construct solutions to a corresponding Schroedinger equation with
time-dependent potential. This is done by a generalization of the Doss approach
to time-dependent potentials. This involves an expectation w.r.t. a complex
scaled Brownian motion. As examples polynomial potentials of degree $4n+2,
n\in\mathbb N,$ and singular potentials of the form $\frac{1}{|x|^n},
n\in\mathbb N$ and $\frac{1}{x^n}, n\in\mathbb N,$ are worked out. | math-ph |
Schrödinger-Koopman quasienergy states of quantum systems driven by a
classical flow: We study the properties of the quasienergy states of a quantum system driven
by a classical dynamical system. The quasienergies are defined in a same manner
as in light-matter interaction but where the Floquet approach is generalized by
the use of the Koopman approach of dynamical systems. We show how the
properties of the classical flow (fixed and cyclic points, ergodicity, chaos)
influence the driven quantum system. This approach of the Schr\"odinger-Koopman
quasienergies can be applied to quantum control, quantum information in
presence of noises, and dynamics of mixed classical-quantum systems. We treat
the example of a kicked spin ensemble where the kick modulation is governed by
discrete classical flows as the Arnold's cat map and the Chirikov standard map. | math-ph |
Symmetries and Casimirs of radial compressible fluid flow and gas
dynamics in n>1 dimensions: Symmetries and Casimirs are studied for the Hamiltonian equations of radial
compressible fluid flow in n>1 dimensions. An explicit determination of all Lie
point symmetries is carried out, from which a complete classification of all
maximal Lie symmetry algebras is obtained. The classification includes all Lie
point symmetries that exist only for special equations of state. For a general
equation of state, the hierarchy of advected conserved integrals found in
recent work is proved to consist of Hamiltonian Casimirs. A second hierarchy
that holds only for an entropic equation of state is explicitly shown to
comprise non-Casimirs which yield a corresponding hierarchy of generalized
symmetries through the Hamiltonian structure of the equations of radial fluid
flow. The first-order symmetries are shown to generate a non-abelian Lie
algebra. Two new kinematic conserved integrals found in recent work are
likewise shown to yield additional first-order generalized symmetries holding
for a barotropic equation of state and an entropic equation of state. These
symmetries produce an explicit transformation group acting on solutions of the
fluid equations. Since these equations are well known to be equivalent to the
equations of gas dynamics, all of the results obtained for n-dimensional radial
fluid flow carry over to radial gas dynamics. | math-ph |
Additional symmetry of the modified extended Toda hierarchy: In this paper, one new integrable modified extended Toda hierarchy(METH) is
constructed with the help of two logarithmic Lax operators. With this
modification, the interpolated spatial flow is added to make all flows
complete. To show more integrable properties of the METH, the bi-Hamiltonian
structure and tau symmetry of the METH will be given. The additional symmetry
flows of this new hierarchy are presented. These flows form an infinite
dimensional Lie algebra of Block type. | math-ph |
Generalized cycles on Spectral Curves: Generalized cycles can be thought of as the extension of form-cycle duality
between holomorphic forms and cycles, to meromorphic forms and generalized
cycles. They appeared as an ubiquitous tool in the study of spectral curves and
integrable systems in the topological recursion approach. They parametrize
deformations, implementing the special geometry, where moduli are periods, and
derivatives with respect to moduli are other periods, or more generally
"integrals", whence the name "generalized cycles". They appeared over the years
in various works, each time in specific applied frameworks, and here we provide
a comprehensive self-contained corpus of definitions and properties for a very
general setting. The geometry of generalized cycles is also fascinating by
itself. | math-ph |
A Combinatorial Description of Certain Polynomials Related to the XYZ
Spin Chain: We study the connection between the three-color model and the polynomials
$q_n(z)$ of Bazhanov and Mangazeev, which appear in the eigenvectors of the
Hamiltonian of the XYZ spin chain. By specializing the parameters in the
partition function of the 8VSOS model with DWBC and reflecting end, we find an
explicit combinatorial expression for $q_n(z)$ in terms of the partition
function of the three-color model with the same boundary conditions. Bazhanov
and Mangazeev conjectured that $q_n(z)$ has positive integer coefficients. We
prove the weaker statement that $q_n(z+1)$ and $(z+1)^{n(n+1)}q_n(1/(z+1))$
have positive integer coefficients. Furthermore, for the three-color model, we
find some results on the number of states with a given number of faces of each
color, and we compute strict bounds for the possible number of faces of each
color. | math-ph |
Theory and application of Fermi pseudo-potential in one dimension: The theory of interaction at one point is developed for the one-dimensional
Schrodinger equation. In analog with the three-dimensional case, the resulting
interaction is referred to as the Fermi pseudo-potential. The dominant feature
of this one-dimensional problem comes from the fact that the real line becomes
disconnected when one point is removed. The general interaction at one point is
found to be the sum of three terms, the well-known delta-function potential and
two Fermi pseudo-potentials, one odd under space reflection and the other even.
The odd one gives the proper interpretation for the delta'(x) potential, while
the even one is unexpected and more interesting. Among the many applications of
these Fermi pseudo-potentials, the simplest one is described. It consists of a
superposition of the delta-function potential and the even pseudo-potential
applied to two-channel scattering. This simplest application leads to a model
of the quantum memory, an essential component of any quantum computer. | math-ph |
Spectral flow argument localizing an odd index pairing: An odd Fredholm module for a given invertible operator on a Hilbert space is
specified by an unbounded so-called Dirac operator with compact resolvent and
bounded commutator with the given invertible. Associated to this is an index
pairing in terms of a Fredholm operator with Noether index. Here it is shown by
a spectral flow argument how this index can be calculated as the signature of a
finite dimensional matrix called the spectral localizer. | math-ph |
Quantum groups, Yang-Baxter maps and quasi-determinants: For any quasi-triangular Hopf algebra, there exists the universal R-matrix,
which satisfies the Yang-Baxter equation. It is known that the adjoint action
of the universal R-matrix on the elements of the tensor square of the algebra
constitutes a quantum Yang-Baxter map, which satisfies the set-theoretic
Yang-Baxter equation. The map has a zero curvature representation among
L-operators defined as images of the universal R-matrix. We find that the zero
curvature representation can be solved by the Gauss decomposition of a product
of L-operators. Thereby obtained a quasi-determinant expression of the quantum
Yang-Baxter map associated with the quantum algebra $U_{q}(gl(n))$. Moreover,
the map is identified with products of quasi-Pl\"{u}cker coordinates over a
matrix composed of the L-operators. We also consider the quasi-classical limit,
where the underlying quantum algebra reduces to a Poisson algebra. The
quasi-determinant expression of the quantum Yang-Baxter map reduces to ratios
of determinants, which give a new expression of a classical Yang-Baxter map. | math-ph |
Oscillations of Degenerate Plasma in Layer with Specular - Accommodative
Boundary Conditions: The linearized problem of plasma oscillations in layer (particularly, in thin
films) in external longitudinal alternating electric field is solved
analytically. Specular - accommodative boundary conditions of electron
reflection from the plasma boundary are considered. Coefficients of continuous
and discrete spectra of the problem are found, and electron distribution
function on the plasma boundary and electric field are expressed in explicit
form. Absorption of energy of electric field in layer is calculated. | math-ph |
Characterization and parameterization of the singular manifold of a
simple 6-6 Stewart platform: This paper presents a study of the characterization of the singular manifold
of the six-degree-of-freedom parallel manipulator commonly known as the Stewart
platform. We consider a platform with base vertices in a circle and for which
the bottom and top plates are related by a rotation and a contraction. It is
shown that in this case the platform is always in a singular configuration and
that the singular manifold can be parameterized by a scalar parameter. | math-ph |
Shallow-water equations with complete Coriolis force: Group Properties
and Similarity Solutions: The group properties of the shallow-water equations with the complete
Coriolis force is the subject of this study. In particular we apply the Lie
theory to classify the system of three nonlinear partial differential equations
according to the admitted Lie point symmetries. For each case of the
classification problem the one-dimensional optimal system is determined. The
results are applied for the derivation of new similarity solutions. | math-ph |
Non-gaussian waves in Seba's billiard: The Seba billiard, a rectangular torus with a point scatterer, is a popular
model to study the transition between integrability and chaos in quantum
systems. Whereas such billiards are classically essentially integrable, they
may display features such as quantum ergodicity [KU] which are usually
associated with quantum systems whose classical dynamics is chaotic. Seba
proposed that the eigenfunctions of toral point scatterers should also satisfy
Berry's random wave conjecture, which implies that the semiclassical moments of
the eigenfunctions ought to be Gaussian.
We prove a conjecture of Keating, Marklof and Winn who suggested that Seba
billiards with irrational aspect ratio violate the random wave conjecture. More
precisely, in the case of diophantine tori, we construct a subsequence of
eigenfunctions of essentially full density and show that its semiclassical
moments cannot be Gaussian. | math-ph |
Extension of Grimus-Stockinger formula from operator expansion of free
Green function: The operator expansion of free Green function of Helmholtz equation for
arbitrary N- dimension space leads to asymptotic extension of 3- dimension
Grimus-Stockinger formula closely related to multipole expansion. Analytical
examples inspired by neutrino oscillation and neutrino deficit problems are
considered for relevant class of wave packets | math-ph |
Stability transitions for axisymmetric relative equilibria of Euclidean
symmetric Hamiltonian systems: In the presence of noncompact symmetry, the stability of relative equilibria
under momentum-preserving perturbations does not generally imply robust
stability under momentum-changing perturbations. For axisymmetric relative
equilibria of Hamiltonian systems with Euclidean symmetry, we investigate
different mechanisms of stability: stability by energy-momentum confinement,
KAM, and Nekhoroshev stability, and we explain the transitions between these.
We apply our results to the Kirchhoff model for the motion of an axisymmetric
underwater vehicle, and we numerically study dissipation induced instability of
KAM stable relative equilibria for this system. | math-ph |
Action of $W$-type operators on Schur functions and Schur Q-functions: In this paper, we investigate a series of W-type differential operators,
which appear naturally in the symmetry algebras of KP and BKP hierarchies. In
particular, they include all operators in the W-constraints for tau functions
of higher KdV hierarchies which satisfy the string equation. We will give
simple uniform formulas for actions of these operators on all ordinary Schur
functions and Schur's Q-functions. As applications of such formulas, we will
give new simple proofs for Alexandrov's conjecture and Mironov-Morozov's
formula, which express the Br\'{e}zin-Gross-Witten and Kontsevich-Witten
tau-functions as linear combinations of Q-functions with simple coefficients
respectively. | math-ph |
The Fermionic Signature Operator in De Sitter Spacetime: The fermionic projector state is a distinguished quasi-free state for the
algebra of Dirac fields in a globally hyperbolic spacetime. We construct and
analyze it in the four-dimensional de Sitter spacetime, both in the closed and
in the flat slicing. In the latter case we show that the mass oscillation
properties do not hold due to boundary effects. This is taken into account in a
so-called mass decomposition. The involved fermionic signature operator defines
a fermionic projector state. In the case of a closed slicing, we construct the
fermionic signature operator and show that the ensuing state is maximally
symmetric and of Hadamard form, thus coinciding with the counterpart for
spinors of the Bunch-Davies state. | math-ph |
Asymptotics of spacing distributions at the hard edge for
$β$-ensembles: In a previous work [J. Math. Phys. {\bf 35} (1994), 2539--2551], generalized
hypergeometric functions have been used to a give a rigorous derivation of the
large $s$ asymptotic form of the general $\beta > 0$ gap probability
$E_\beta^{\rm hard}(0;(0,s);\beta a/2)$, provided both $\beta a /2 \in \mathbb
Z_\ge 0$ and $2/\beta \in \mathbb Z^+$. It shown how the details of this method
can be extended to remove the requirement that $2/\beta \in \mathbb Z^+$.
Furthermore, a large deviation formula for the gap probability
$E_\beta(n;(0,x);{\rm ME}_{\beta,N}(\lambda^{a \beta /2} e^{\beta N
\lambda/2}))$ is deduced by writing it in terms of the charateristic function
of a certain linear statistic. By scaling $x = s/(4N)^2$ and taking $N \to
\infty$, this is shown to reproduce a recent conjectured formula for
$E_\beta^{\rm hard}(n;(0,s);\beta a/2)$, $\beta a /2 \in \mathbb Z_{\ge 0}$,
and moreover to give a prediction without the latter restriction. This extended
formula, which for the constant term involves the Barnes double gamma function,
is shown to satisfy an asymptotic functional equation relating the gap
probability with parameters $(\beta,n,a)$, to a gap probability with parameters
$(4/\beta,n',a')$, where $n'=\beta(n+1)/2-1$, $a'=\beta(a-2)/2+2$. | math-ph |
Absolute convergence of the free energy of the BEG model in the
disordered region for all temperatures: We analyze the d-dimensional Blume-Emery-Griffiths model in the disordered
region of parameters and we show that its free energy can be explicitly written
in term of a series which is absolutely convergent at any temperature in an
unbounded portion of this region. As a byproduct we also obtain an upper bound
for the number of d-dimensional fixed polycubes of size n. | math-ph |
On an elastic strain-limiting special Cosserat rod model: Motivated by recent strain-limiting models for solids and biological fibers,
we introduce the first intrinsic set of nonlinear constitutive relations,
between the geometrically exact strains and the components of the contact force
and contact couple, describing a uniform, hyperelastic, strain-limiting special
Cosserat rod. After discussing some attractive features of the constitutive
relations (orientation preservation, transverse symmetry, and monotonicity), we
exhibit several explicit equilibrium states under either an isolated end thrust
or an isolated end couple. In particular, certain equilibrium states exhibit
Poynting like effects, and we show that under mild assumptions on the material
parameters, the model predicts an explicit tensile shearing bifurcation: a
straight rod under a large enough tensile end thrust parallel to its center
line can shear. | math-ph |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.