text
stringlengths 73
2.82k
| category
stringclasses 21
values |
|---|---|
Eigenvalue crossings in Floquet topological systems: The topology of electrons on a lattice subject to a periodic driving is
captured by the three-dimensional winding number of the propagator that
describes time-evolution within a cycle. This index captures the homotopy class
of such a unitary map. In this paper, we provide an interpretation of this
winding number in terms of local data associated to the the eigenvalue
crossings of such a map over a three dimensional manifold, based on an idea
from Nathan and Rudner, New Journal of Physics, 17(12) 125014, 2015. We show
that, up to homotopy, the crossings are a finite set of points and non
degenerate. Each crossing carries a local Chern number, and the sum of these
local indices coincides with the winding number. We then extend this result to
fully degenerate crossings and extended submanifolds to connect with models
from the physics literature. We finally classify up to homotopy the Floquet
unitary maps, defined on manifolds with boundary, using the previous local
indices. The results rely on a filtration of the special unitary group as well
as the local data of the basic gerbe over it. | math-ph |
Chain of matrices, loop equations and topological recursion: Random matrices are used in fields as different as the study of
multi-orthogonal polynomials or the enumeration of discrete surfaces. Both of
them are based on the study of a matrix integral. However, this term can be
confusing since the definition of a matrix integral in these two applications
is not the same. These two definitions, perturbative and non-perturbative, are
discussed in this chapter as well as their relation. The so-called loop
equations satisfied by integrals over random matrices coupled in chain is
discussed as well as their recursive solution in the perturbative case when the
matrices are Hermitean. | math-ph |
Dirac structures in nonequilibrium thermodynamics: Dirac structures are geometric objects that generalize both Poisson
structures and presymplectic structures on manifolds. They naturally appear in
the formulation of constrained mechanical systems. In this paper, we show that
the evolution equa- tions for nonequilibrium thermodynamics admit an intrinsic
formulation in terms of Dirac structures, both on the Lagrangian and the
Hamiltonian settings. In absence of irreversible processes these Dirac
structures reduce to canonical Dirac structures associated to canonical
symplectic forms on phase spaces. Our geometric formulation of nonequilibrium
thermodynamic thus consistently extends the geometric formulation of mechanics,
to which it reduces in absence of irreversible processes. The Dirac structures
are associated to the variational formulation of nonequilibrium thermodynamics
developed in Gay-Balmaz and Yoshimura [2016a,b] and are induced from a
nonlinear nonholonomic constraint given by the expression of the entropy
production of the system. | math-ph |
Bethe ansatz and Hirota equation in integrable models: In this short review the role of the Hirota equation and the tau-function in
the theory of classical and quantum integrable systems is outlined. | math-ph |
Combined mean-field and semiclassical limits of large fermionic systems: We study the time dependent Schr\"odinger equation for large spinless
fermions with the semiclassical scale $\hbar = N^{-1/3}$ in three dimensions.
By using the Husimi measure defined by coherent states, we rewrite the
Schr\"odinger equation into a BBGKY type of hierarchy for the k particle Husimi
measure. Further estimates are derived to obtain the weak compactness of the
Husimi measure, and in addition uniform estimates for the remainder terms in
the hierarchy are derived in order to show that in the semiclassical regime the
weak limit of the Husimi measure is exactly the solution of the Vlasov
equation. | math-ph |
Effective Hamiltonians for atoms in very strong magnetic fields: We propose three effective Hamiltonians which approximate atoms in very
strong homogeneous magnetic fields $B$ modelled by the Pauli Hamiltonian, with
fixed total angular momentum with respect to magnetic field axis. All three
Hamiltonians describe $N$ electrons and a fixed nucleus where the Coulomb
interaction has been replaced by $B$-dependent one-dimensional effective
(vector valued) potentials but without magnetic field. Two of them are solvable
in at least the one electron case. We briefly sketch how these Hamiltonians can
be used to analyse the bottom of the spectrum of such atoms. | math-ph |
Spectral Curve of the Halphen Operator: The Halphen operator is a third-order operator of the form $$
L_3=\partial_x^3-g(g+2)\wp(x)\partial_x-\frac{1}{2}g(g+2)\wp'(x), $$ where
$g\ne 2\,\mbox{mod(3)}$, the Weierstrass $\wp$-function satisfies the equation
$$
(\wp'(x))^2=4\wp^3(x)-g_2\wp(x)-g_3. $$ In the equianharmonic case, i.e.,
$g_2=0$ the Halphen operator commutes with some ordinary differential operator
$L_n$ of order $n\ne 0\,\mbox{mod(3)}.$ In this paper we find the spectral
curve of the pair $L_3,L_n$. | math-ph |
Proof of the orthogonal--Pin duality: This article contains the proof of a theorem on orthogonal-Pin duality that
was cited without proof in a previous article in this journal. | math-ph |
Spherical functions on the de Sitter group: Matrix elements and spherical functions of irreducible representations of the
de Sitter group are studied on the various homogeneous spaces of this group. It
is shown that a universal covering of the de Sitter group gives rise to
quaternion Euler angles. An explicit form of Casimir and Laplace-Beltrami
operators on the homogeneous spaces is given. Different expressions of the
matrix elements and spherical functions are given in terms of multiple
hypergeometric functions both for finite-dimensional and unitary
representations of the principal series of the de Sitter group. | math-ph |
Universality of the Hall conductivity in interacting electron systems: We prove the quantization of the Hall conductivity for general weakly
interacting gapped fermionic systems on two-dimensional periodic lattices. The
proof is based on fermionic cluster expansion techniques combined with lattice
Ward identities, and on a reconstruction theorem that allows us to compute the
Kubo conductivity as the analytic continuation of its imaginary time
counterpart. | math-ph |
On energy-momentum transfer of quantum fields: We prove the following theorem on bounded operators in quantum field theory:
if $\|[B,B^*(x)]\|\leq \mathrm{const} D(x)$, then
$\|B^k_\pm(\nu)G(P^0)\|^2\leq\mathrm{const}\int D(x-y)d|\nu|(x)d|\nu|(y)$,
where $D(x)$ is a function weakly decaying in spacelike directions, $B^k_\pm$
are creation/annihilation parts of an appropriate time derivative of $B$, $G$
is any positive, bounded, non-increasing function in $L^2(\mathbb{R})$, and
$\nu$ is any finite complex Borel measure; creation/annihilation operators may
be also replaced by $B^k_t$ with $\check{B^k_t}(p)=|p|^k\check{B}(p)$. We also
use the notion of energy-momentum scaling degree of $B$ with respect to a
submanifold (Steinmann-type, but in momentum space, and applied to the norm of
an operator). These two tools are applied to the analysis of singularities of
$\check{B}(p)G(P^0)$. We prove, among others, the following statement (modulo
some more specific assumptions): outside $p=0$ the only allowed contributions
to this functional which are concentrated on a submanifold (including the
trivial one -- a single point) are Dirac measures on hypersurfaces (if the
decay of $D$ is not to slow). | math-ph |
Generalized point vortex dynamics on $CP ^2$: This is the second of two companion papers. We describe a generalization of
the point vortex system on surfaces to a Hamiltonian dynamical system
consisting of two or three points on complex projective space CP^2 interacting
via a Hamiltonian function depending only on the distance between the points.
The system has symmetry group SU(3). The first paper describes all possible
momentum values for such systems, and here we apply methods of symplectic
reduction and geometric mechanics to analyze the possible relative equilibria
of such interacting generalized vortices.
The different types of polytope depend on the values of the `vortex
strengths', which are manifested as coefficients of the symplectic forms on the
copies of CP^2. We show that the reduced space for this Hamiltonian action for
3 vortices is generically a 2-sphere, and proceed to describe the reduced
dynamics under simple hypotheses on the type of Hamiltonian interaction. The
other non-trivial reduced spaces are topological spheres with isolated singular
points. For 2 generalized vortices, the reduced spaces are just points, and the
motion is governed by a collective Hamiltonian, whereas for 3 the reduced
spaces are of dimension at most 2. In both cases the system will be completely
integrable in the non-abelian sense. | math-ph |
An exactly-solvable three-dimensional nonlinear quantum oscillator: Exact analytical, closed-form solutions, expressed in terms of special
functions, are presented for the case of a three-dimensional nonlinear quantum
oscillator with a position dependent mass. This system is the generalization of
the corresponding one-dimensional system, which has been the focus of recent
attention. In contrast to other approaches, we are able to obtain solutions in
terms of special functions, without a reliance upon a Rodrigues-type of
formula. The wave functions of the quantum oscillator have the familiar
spherical harmonic solutions for the angular part. For the s-states of the
system, the radial equation accepts solutions that have been recently found for
the one-dimensional nonlinear quantum oscillator, given in terms of associated
Legendre functions, along with a constant shift in the energy eigenvalues.
Radial solutions are obtained for all angular momentum states, along with the
complete energy spectrum of the bound states. | math-ph |
The Coleman correspondence at the free fermion point: We prove that the truncated correlation functions of the charge and gradient
fields associated with the massless sine-Gordon model on $\mathbb{R}^2$ with
$\beta=4\pi$ exist for all coupling constants and are equal to those of the
chiral densities and vector current of free massive Dirac fermions. This is an
instance of Coleman's prediction that the massless sine-Gordon model and the
massive Thirring model are equivalent (in the above sense of correlation
functions). Our main novelty is that we prove this correspondence starting from
the Euclidean path integral in the non-perturative regime of the infinite
volume models. We use this correspondence to show that the correlation
functions of the massless sine-Gordon model with $\beta=4\pi$ decay
exponentially and that the corresponding probabilistic field is localized. | math-ph |
On the Spectrum of Holonomy Algebras: The paper has been withdrawn by the authors | math-ph |
Duality properties of Gorringe-Leach equations: In the category of motions preserving the angular momentum's direction,
Gorringe and Leach exhibited two classes of differential equations having
elliptical orbits. After enlarging slightly these classes, we show that they
are related by a duality correspondence of the Arnold-Vassiliev type. The
specific associated conserved quantities (Laplace-Runge-Lenz vector and
Fradkin-Jauch-Hill tensor) are then dual reflections one of the other | math-ph |
An Exterior Algebraic Derivation of the Euler-Lagrange Equations from
the Principle of Stationary Action: In this paper, we review two related aspects of field theory: the modeling of
the fields by means of exterior algebra and calculus, and the derivation of the
field dynamics, i.e., the Euler-Lagrange equations, by means of the stationary
action principle. In contrast to the usual tensorial derivation of these
equations for field theories, that gives separate equations for the field
components, two related coordinate-free forms of the Euler-Lagrange equations
are derived. These alternative forms of the equations, reminiscent of the
formulae of vector calculus, are expressed in terms of vector derivatives of
the Lagrangian density. The first form is valid for a generic Lagrangian
density that only depends on the first-order derivatives of the field. The
second form, expressed in exterior algebra notation, is specific to the case
when the Lagrangian density is a function of the exterior and interior
derivatives of the multivector field. As an application, a Lagrangian density
for generalized electromagnetic multivector fields of arbitrary grade is
postulated and shown to have, by taking the vector derivative of the Lagrangian
density, the generalized Maxwell equations as Euler--Lagrange equations. | math-ph |
On Dynamical Justification of Quantum Scattering Cross Section: A~dynamical justification of quantum differential cross section in the
context of long time transition to stationary regime for the Schr\"odinger
equation is suggested. The problem has been stated by Reed and Simon. Our
approach is based on spherical incident waves produced by a harmonic source and
the long-range asymptotics for the corresponding spherical limiting amplitudes.
The main results are as follows: i)~the convergence of spherical limiting
amplitudes to the limit as the source increases to infinity, and ii) the
universally recognized formula for the differential cross section corresponding
to the limiting flux. The main technical ingredients are the
Agmon--Jensen--Kato's analytical theory of the Green function, Ikebe's
uniqueness theorem for the Lippmann--Schwinger equation, and some adjustments
of classical asymptotics for the Coulomb potentials. | math-ph |
Duality family of KdV equation: It is revealed that there exist duality families of the KdV type equation. A
duality family consists of an infinite number of generalized KdV (GKdV)
equations. A duality transformation relates the GKdV equations in a duality
family. Once a family member is solved, the duality transformation presents the
solutions of all other family members. We show some dualities as examples, such
as the soliton solution-soliton solution duality and the periodic
solution-soliton solution duality. | math-ph |
Converging Perturbative Solutions of the Schroedinger Equation for a
Two-Level System with a Hamiltonian Depending Periodically on Time: We study the Schroedinger equation of a class of two-level systems under the
action of a periodic time-dependent external field in the situation where the
energy difference 2epsilon between the free energy levels is sufficiently small
with respect to the strength of the external interaction. Under suitable
conditions we show that this equation has a solution in terms of converging
power series expansions in epsilon. In contrast to other expansion methods,
like in the Dyson expansion, the method we present is not plagued by the
presence of ``secular terms''. Due to this feature we were able to prove
absolute and uniform convergence of the Fourier series involved in the
computation of the wave functions and to prove absolute convergence of the
epsilon-expansions leading to the ``secular frequency'' and to the coefficients
of the Fourier expansion of the wave function. | math-ph |
Fermion Quasi-Spherical Harmonics: Spherical Harmonics, $Y_\ell^m(\theta,\phi)$, are derived and presented (in a
Table) for half-odd-integer values of $\ell$ and $m$. These functions are
eigenfunctions of $L^2$ and $L_z$ written as differential operators in the
spherical-polar angles, $\theta$ and $\phi$. The Fermion Spherical Harmonics
are a new, scalar and angular-coordinate-dependent representation of fermion
spin angular momentum. They have $4\pi$ symmetry in the angle $\phi$, and hence
are not single-valued functions on the Euclidean unit sphere; they are
double-valued functions on the sphere, or alternatively are interpreted as
having a double-sphere as their domain. | math-ph |
Approximate Q-conditional symmetries of partial differential equations: Following a recently introduced approach to approximate Lie symmetries of
differential equations which is consistent with the principles of perturbative
analysis of differential equations containing small terms, we analyze the case
of approximate $Q$--conditional symmetries. An application of the method to a
hyperbolic variant of a reaction--diffusion--convection equation is presented. | math-ph |
Ground state and orbital stability for the NLS equation on a general
starlike graph with potentials: We consider a nonlinear Schr\"odinger equation (NLS) posed on a graph or
network composed of a generic compact part to which a finite number of
half-lines are attached. We call this structure a starlike graph. At the
vertices of the graph interactions of $\delta$-type can be present and an
overall external potential is admitted. Under general assumptions on the
potential, we prove that the NLS is globally well-posed in the energy domain.
We are interested in minimizing the energy of the system on the manifold of
constant mass ($L^2$-norm). When existing, the minimizer is called ground state
and it is the profile of an orbitally stable standing wave for the NLS
evolution. We prove that a ground state exists for sufficiently small masses
whenever the quadratic part of the energy admits a simple isolated eigenvalue
at the bottom of the spectrum (the linear ground state). This is a wide
generalization of a result previously obtained for a star graph with a single
vertex. The main part of the proof is devoted to prove the concentration
compactness principle for starlike structures; this is non trivial due to the
lack of translation invariance of the domain. Then we show that a minimizing
bounded $H^1$ sequence for the constrained NLS energy with external linear
potentials is in fact convergent if its mass is small enough. Examples are
provided with discussion of hypotheses on the linear part. | math-ph |
Resonances for 1D massless Dirac operators: We consider the 1D massless Dirac operator on the real line with compactly
supported potentials. We study resonances as the poles of scattering matrix or
equivalently as the zeros of modified Fredholm determinant. We obtain the
following properties of the resonances: 1) asymptotics of counting function, 2)
estimates on the resonances and the forbidden domain, 3) the trace formula in
terms of resonances. | math-ph |
Invariant classification of second-order conformally flat
superintegrable systems: In this paper we continue the work of Kalnins et al in classifying all
second-order conformally-superintegrable (Laplace-type) systems over
conformally flat spaces, using tools from algebraic geometry and classical
invariant theory. The results obtained show, through Staeckel equivalence, that
the list of known nondegenerate superintegrable systems over three-dimensional
conformally flat spaces is complete. In particular, a 7-dimensional manifold is
determined such that each point corresponds to a conformal class of
superintegrable systems. This manifold is foliated by the nonlinear action of
the conformal group in three-dimensions. Two systems lie in the same conformal
class if and only if they lie in the same leaf of the foliation. This foliation
is explicitly described using algebraic varieties formed from representations
of the conformal group. The proof of these results rely heavily on Groebner
basis calculations using the computer algebra software packages Maple and
Singular. | math-ph |
A unifying perspective on linear continuum equations prevalent in
science. Part II: Canonical forms for time-harmonic equations: Following some past advances, we reformulate a large class of linear
continuum science equations in the format of the extended abstract theory of
composites so that we can apply this theory to better understand and
efficiently solve those equations. Here in part II we elucidate the form for
many time-harmonic equations that do not involve higher order gradients. | math-ph |
The Berry phase and the phase of the determinant: In 1984 Michael Berry discovered that an isolated eigenstate of an
adiabatically changing periodic Hamiltonian $H(t)$ acquires a phase, called the
Berry phase. We show that under very general assumptions the adiabatic
approximation of the phase of the zeta-regularized determinant of the
imaginary-time Schrodinger operator with periodic Hamiltonian is equal to the
Berry phase. | math-ph |
On the reality of spectra of $\boldsymbol{U_q(sl_2)}$-invariant XXZ
Hamiltonians: A new inner product is constructed on each standard module over the
Temperley-Lieb algebra $\mathsf{TL}_n(\beta)$ for $\beta\in \mathbb R$ and $n
\ge 2$. On these modules, the Hamiltonian $h = -\sum_i e_i$ is shown to be
self-adjoint with respect to this inner product. This implies that its action
on these modules is diagonalisable with real eigenvalues. A representation
theoretic argument shows that the reality of spectra of the Hamiltonian extends
to all other Temperley-Lieb representations. In particular, this result applies
to the celebrated $U_q(sl_2)$-invariant XXZ Hamiltonian, for all $q+q^{-1}\in
\mathbb R$. | math-ph |
Deformation of the J-matrix method of scattering: We construct nonrelativistic J-matrix theory of scattering for a system whose
reference Hamiltonian is enhanced by one-parameter linear deformation to
account for nontrivial physical effects that could be modeled by a singular
ground state coupling. | math-ph |
Chiral Asymmetry and the Spectral Action: We consider orthogonal connections with arbitrary torsion on compact
Riemannian manifolds. For the induced Dirac operators, twisted Dirac operators
and Dirac operators of Chamseddine-Connes type we compute the spectral action.
In addition to the Einstein-Hilbert action and the bosonic part of the Standard
Model Lagrangian we find the Holst term from Loop Quantum Gravity, a coupling
of the Holst term to the scalar curvature and a prediction for the value of the
Barbero-Immirzi parameter. | math-ph |
Nonexistence of steady solutions for rotational slender fibre spinning
with surface tension: Reduced one-dimensional equations for the stationary, isothermal rotational
spinning process of slender fibers are considered for the case of large
Reynolds ($\delta=3/\text{Re}\ll 1$) and small Rossby numbers ($\varepsilon \ll
1$). Surface tension is included in the model using the parameter
$\kappa=\sqrt{\pi}/(2 \text{We})$ related to the inverse Weber number. The
inviscid case $\delta=0$ is discussed as a reference case. For the viscous case
$\delta > 0$ numerical simulations indicate, that for a certain parameter
range, no physically relevant solution may exist. Transferring properties of
the inviscid limit to the viscous case, analytical bounds for the initial
viscous stress of the fiber are obtained. A good agreement with the numerical
results is found. These bounds give strong evidence, that for $\delta >
3\varepsilon^2 \left( 1- \frac{3}{2}\kappa +\frac{1}{2}\kappa^2\right)$ no
physical relevant stationary solution can exist. | math-ph |
The Hartree-von Neumann limit of many body dynamics: In the mean-field regime, we prove convergence (with explicit bounds) of the
many-body von Neumann dynamics with bounded interactions to the Hartree-von
Neumann dynamics. | math-ph |
Localization Properties of the Chalker-Coddington Model: The Chalker Coddington quantum network percolation model is numerically
pertinent to the understanding of the delocalization transition of the quantum
Hall effect. We study the model restricted to a cylinder of perimeter 2M. We
prove firstly that the Lyapunov exponents are simple and in particular that the
localization length is finite; secondly that this implies spectral
localization. Thirdly we prove a Thouless formula and compute the mean Lyapunov
exponent which is independent of M. | math-ph |
Generalized Laguerre Polynomials with Position-Dependent Effective Mass
Visualized via Wigner's Distribution Functions: We construct, analytically and numerically, the Wigner distribution functions
for the exact solutions of position-dependent effective mass Schr\"odinger
equation for two cases belonging to the generalized Laguerre polynomials. Using
a suitable quantum canonical transformation, expectation values of position and
momentum operators can be obtained analytically in order to verify the
universality of the Heisenberg's uncertainty principle. | math-ph |
The Yang-Baxter equation for PT invariant nineteen vertex models: We study the solutions of the Yang-Baxter equation associated to nineteen
vertex models invariant by the parity-time symmetry from the perspective of
algebraic geometry. We determine the form of the algebraic curves constraining
the respective Boltzmann weights and found that they possess a universal
structure. This allows us to classify the integrable manifolds in four
different families reproducing three known models besides uncovering a novel
nineteen vertex model in a unified way. The introduction of the spectral
parameter on the weights is made via the parameterization of the fundamental
algebraic curve which is a conic. The diagonalization of the transfer matrix of
the new vertex model and its thermodynamic limit properties are discussed. We
point out a connection between the form of the main curve and the nature of the
excitations of the corresponding spin-1 chains. | math-ph |
New Bessel Identities from Laguerre Polynomials: For large order, Laguerre polynomials can be approximated by Bessel functions
near the origin. This can be used to turn many Laguerre identities into
corresponding identities for Bessel functions. We will illustrate this idea
with a number of examples. In particular, we will derive a generalization of a
identity due to Sonine, which appears to be new. | math-ph |
New self-dual solutions of SU(2) Yang-Mills theory in Euclidean
Schwarzschild space: We present a systematic study of spherically symmetric self-dual solutions of
SU(2) Yang-Mills theory on Euclidean Schwarzschild space. All the previously
known solutions are recovered and a new one-parameter family of instantons is
obtained. The newly found solutions have continuous actions and interpolate
between the classic Charap and Duff instantons. We examine the physical
properties of this family and show that it consists of dyons of unit (magnetic
and electric) charge. | math-ph |
The wave equation on singular space-times: We prove local unique solvability of the wave equation for a large class of
weakly singular, locally bounded space-time metrics in a suitable space of
generalised functions. | math-ph |
Instabilities Appearing in Cosmological Effective Field theories: When
and How?: Nonlinear partial differential equations appear in many domains of physics,
and we study here a typical equation which one finds in effective field
theories (EFT) originated from cosmological studies. In particular, we are
interested in the equation $\partial_t^2 u(x,t) = \alpha (\partial_x u(x,t))^2
+\beta \partial_x^2 u(x,t)$ in $1+1$ dimensions. It has been known for quite
some time that solutions to this equation diverge in finite time, when $\alpha
>0$. We study the nature of this divergence as a function of the parameters
$\alpha>0 $ and $\beta\ge0$. The divergence does not disappear even when $\beta
$ is very large contrary to what one might believe (note that since we consider
fixed initial data, $\alpha$ and $\beta$ cannot be scaled away). But it will
take longer to appear as $\beta$ increases when $\alpha$ is fixed. We note that
there are two types of divergence and we discuss the transition between these
two as a function of parameter choices. The blowup is unavoidable unless the
corresponding equations are modified. Our results extend to $3+1$ dimensions. | math-ph |
Variations on a theme of q-oscillator: We present several ideas in direction of physical interpretation of $q$- and
$f$-oscillators as a nonlinear oscillators. First we show that an arbitrary one
dimensional integrable system in action-angle variables can be naturally
represented as a classical and quantum $f$-oscillator. As an example, the
semi-relativistic oscillator as a descriptive of the Landau levels for
relativistic electron in magnetic field is solved as an $f$-oscillator. By
using dispersion relation for $q$-oscillator we solve the linear
q-Schr\"odinger equation and corresponding nonlinear complex q-Burgers
equation. The same dispersion allows us to construct integrable q-NLS model as
a deformation of cubic NLS in terms of recursion operator of NLS hierarchy.
Peculiar property of the model is to be completely integrable at any order of
expansion in deformation parameter around $q=1$. As another variation on the
theme, we consider hydrodynamic flow in bounded domain. For the flow bounded by
two concentric circles we formulate the two circle theorem and construct
solution as the q-periodic flow by non-symmetric $q$-calculus. Then we
generalize this theorem to the flow in the wedge domain bounded by two arcs.
This two circular-wedge theorem determines images of the flow by extension of
$q$-calculus to two bases: the real one, corresponding to circular arcs and the
complex one, with $q$ as a primitive root of unity. As an application, the
vortex motion in annular domain as a nonlinear oscillator in the form of
classical and quantum f-oscillator is studied. Extending idea of q-oscillator
to two bases with the golden ratio, we describe Fibonacci numbers as a special
type of $q$-numbers with matrix Binet formula. We derive the corresponding
golden quantum oscillator, nonlinear coherent states and Fock-Bargman
representation. | math-ph |
The electron densities of pseudorelativistic eigenfunctions are smooth
away from the nuclei: We consider a pseudorelativistic model of atoms and molecules, where the
kinetic energy of the electrons is given by $\sqrt{p^2+m^2}-m$. In this model
the eigenfunctions are generally not even bounded, however, we prove that the
corresponding one-electron densities are smooth away from the nuclei. | math-ph |
Two-term asymptotics of the exchange energy of the electron gas on
symmetric polytopes in the high-density limit: We derive a two-term asymptotic expansion for the exchange energy of the free
electron gas on strictly tessellating polytopes and fundamental domains of
lattices in the thermodynamic limit. This expansion comprises a bulk
(volume-dependent) term, the celebrated Dirac exchange, and a novel surface
correction stemming from a boundary layer and finite-size effects. Furthermore,
we derive analogous two-term asymptotic expansions for semi-local density
functionals. By matching the coefficients of these asymptotic expansions, we
obtain an integral constraint for semi-local approximations of the exchange
energy used in density functional theory. | math-ph |
Hardy space on the polydisk and scattering in layered media: Hardy space on the polydisk provides the setting for a global description of
scattering in piecewise-constant layered media, giving a simple qualitative
interpretation for the nonlinear dependence of the Green's function on
reflection coefficients and layer depths. Using explicit formulas for
amplitudes, we prove that the power spectrum of the Green's function is
approximately constant. In addition we exploit a connection to Jacobi
polynomials to derive formulas for computing reflection coefficients from
partial amplitude data. Unlike most approaches to layered media, which
variously involve scaling limits, approximations or iterative methods, the
formulas and methods in the present paper are exact and direct. | math-ph |
Tsallis entropy and generalized Shannon additivity: The Tsallis entropy given for a positive parameter $\alpha$ can be considered
as a modification of the classical Shannon entropy. For the latter,
corresponding to $\alpha=1$, there exist many axiomatic characterizations. One
of them based on the well-known Khinchin-Shannon axioms has been simplified
several times and adapted to Tsallis entropy, where the axiom of (generalized)
Shannon additivity is playing a central role. The main aim of this paper is to
discuss this axiom in the context of Tsallis entropy. We show that it is
sufficient for characterizing Tsallis entropy with the exceptions of cases
$\alpha=1,2$ discussed separately. | math-ph |
The Schrödinger Equation with a Moving Point Interaction in Three
Dimensions: In the case of a single point interaction we improve, by different
techniques, the existence theorem for the unitary evolution generated by a
Schr\"odinger operator with moving point interactions obtained by Dell'Antonio,
Figari and Teta. | math-ph |
Symmetry group analysis of an ideal plastic flow: In this paper, we study the Lie point symmetry group of a system describing
an ideal plastic plane flow in two dimensions in order to find analytical
solutions. The infinitesimal generators that span the Lie algebra for this
system are obtained. We completely classify the subalgebras of up to
codimension two in conjugacy classes under the action of the symmetry group.
Based on invariant forms, we use Ansatzes to compute symmetry reductions in
such a way that the obtained solutions cover simultaneously many invariant and
partially invariant solutions. We calculate solutions of the algebraic,
trigonometric, inverse trigonometric and elliptic type. Some solutions
depending on one or two arbitrary functions of one variable have also been
found. In some cases, the shape of a potentially feasible extrusion die
corresponding to the solution is deduced. These tools could be used to thin,
curve, undulate or shape a ring in an ideal plastic material. | math-ph |
Semi-classical quantization rules for a periodic orbit of hyperbolic
type: Determination of periodic orbits for a Hamiltonian system together with their
semi-classical quantization has been a long standing problem. We consider here
resonances for a $h$-Pseudo-Differential Operator $H(y,hD_y;h)$ induced by a
periodic orbit of hyperbolic type at energy $E_0$. We generalize the framework
of [G\'eSj], in the sense that we allow for both hyperbolic and elliptic
eigenvalues of Poincar\'e map, and show that all resonances in
$W=[E_0-\varepsilon_0,E_0+\varepsilon_0]-i]0,h^\delta]$, $0<\delta<1$, are
given by a generalized Bohr-Sommerfeld quantization rule. | math-ph |
Mechanics of the Infinitesimal Gyroscopes on the Mylar Balloons and
Their Action-Angle Analysis: Here we apply the general scheme for description of the mechanics of
infinitesimal bodies in the Riemannian spaces to the examples of geodetic and
non-geodetic (for two different model potentials) motions of infinitesimal
rotators on the Mylar balloons. The structure of partial degeneracy is
investigated with the help of the corresponding Hamilton-Jacobi equation and
action-angle analysis. In all situations it was found that for any of the six
disjoint regions in the phase space among the three action variables only two
of them are essential for the description of our models at the level of the old
quantum theory (according to the Bohr-Sommerfeld postulates). Moreover, in both
non-geodetic models the action variables were intertwined with the quantum
number $N$ corresponding to the quantization of the radii $r$ of the inflated
Mylar balloons. | math-ph |
A note on normal matrix ensembles at the hard edge: We investigate how the theory of quasipolynomials due to Hedenmalm and
Wennman works in a hard edge setting and obtain as a consequence a scaling
limit for radially symmetric potentials. | math-ph |
Quantum Optimal Transport: Quantum Couplings and Many-Body Problems: This text is a set of lecture notes for a 4.5-hour course given at the
Erd\"os Center (R\'enyi Institute, Budapest) during the Summer School "Optimal
Transport on Quantum Structures" (September 19th-23rd, 2023). Lecture I
introduces the quantum analogue of the Wasserstein distance of exponent $2$
defined in [F. Golse, C. Mouhot, T. Paul: Comm. Math. Phys. 343 (2016),
165-205], and in [F. Golse, T. Paul: Arch. Ration. Mech. Anal. 223 (2017)
57-94]. Lecture II discusses various applications of this quantum analogue of
the Wasserstein distance of exponent $2$, while Lecture III discusses several
of its most important properties, such as the triangle inequality, and the
Kantorovich duality in the quantum setting, together with some of their
implications. | math-ph |
Higher spin sl_2 R-matrix from equivariant (co)homology: We compute the rational $\mathfrak{sl}_2$ $R$-matrix acting in the product of
two spin-$\ell\over 2$ (${\ell \in \mathbb{N}}$) representations, using a
method analogous to the one of Maulik and Okounkov, i.e., by studying the
equivariant (co)homology of certain algebraic varieties. These varieties, first
considered by Nekrasov and Shatashvili, are typically singular. They may be
thought of as the higher spin generalizations of $A_1$ Nakajima quiver
varieties (i.e., cotangent bundles of Grassmannians), the latter corresponding
to $\ell=1$. | math-ph |
The locally covariant Dirac field: We describe the free Dirac field in a four dimensional spacetime as a locally
covariant quantum field theory in the sense of Brunetti, Fredenhagen and Verch,
using a representation independent construction. The freedom in the geometric
constructions involved can be encoded in terms of the cohomology of the
category of spin spacetimes. If we restrict ourselves to the observable algebra
the cohomological obstructions vanish and the theory is unique. We establish
some basic properties of the theory and discuss the class of Hadamard states,
filling some technical gaps in the literature. Finally we show that the
relative Cauchy evolution yields commutators with the stress-energy-momentum
tensor, as in the scalar field case. | math-ph |
On Angles Whose Squared Trigonometric Functions are Rational: We consider the rational linear relations between real numbers whose squared
trigonometric functions have rational values, angles we call ``geodetic''. We
construct a convenient basis for the vector space over Q generated by these
angles. Geodetic angles and rational linear combinations of geodetic angles
appear naturally in Euclidean geometry; for illustration we apply our results
to equidecomposability of polyhedra. | math-ph |
Freud's Identity of Differential Geometry, the Einstein-Hilbert
Equations and the Vexatious Problem of the Energy-Momentum Conservation in GR: We reveal in a rigorous mathematical way using the theory of differential
forms, here viewed as sections of a Clifford bundle over a Lorentzian manifold,
the true meaning of Freud's identity of differential geometry discovered in
1939 (as a generalization of results already obtained by Einstein in 1916) and
rediscovered in disguised forms by several people. We show moreover that
contrary to some claims in the literature there is not a single (mathematical)
inconsistency between Freud's identity (which is a decomposition of the
Einstein indexed 3-forms in two gauge dependent objects) and the field
equations of General Relativity. However, as we show there is an obvious
inconsistency in the way that Freud's identity is usually applied in the
formulation of energy-momentum "conservation laws" in GR. In order for this
paper to be useful for a large class of readers (even those ones making a first
contact with the theory of differential forms) all calculations are done with
all details (disclosing some of the "tricks of the trade" of the subject). | math-ph |
A note on the Baker--Campbell--Hausdorff series in terms of right-nested
commutators: We get compact expressions for the Baker--Campbell--Hausdorff series $Z =
\log(\e^X \, \e^Y)$ in terms of right-nested commutators. The reduction in the
number of terms originates from two facts: (i) we use as a starting point an
explicit expression directly involving independent commutators and (ii) we
derive a complete set of identities arising among right-nested commutators. The
procedure allows us to obtain the series with fewer terms than when expressed
in the classical Hall basis at least up to terms of grade 10. | math-ph |
A particle approximation for the relativistic Vlasov-Maxwell dynamics: We present a microscopic derivation of the 3-dimensional relativistic
Vlasov-Maxwell system as a combined mean field and point-particle limit of an
$N$-particle system of rigid charges with $N$-dependent radius. The
approximation holds for typical initial particle configurations, implying in
particular propagation of chaos for the respective dynamics. | math-ph |
Normal completely positive maps on the space of quantum operations: Quantum supermaps are higher-order maps transforming quantum operations into
quantum operations. Here we extend the theory of quantum supermaps, originally
formulated in the finite dimensional setting, to the case of higher-order maps
transforming quantum operations with input in a separable von Neumann algebra
and output in the algebra of the bounded operators on a given separable Hilbert
space. In this setting we prove two dilation theorems for quantum supermaps
that are the analogues of the Stinespring and Radon-Nikodym theorems for
quantum operations. Finally, we consider the case of quantum superinstruments,
namely measures with values in the set of quantum supermaps, and derive a
dilation theorem for them that is analogue to Ozawa's theorem for quantum
instruments. The three dilation theorems presented here show that all the
supermaps defined in this paper can be implemented by connecting devices in
quantum circuits. | math-ph |
Exact solutions to non-linear classical field theories: We consider some non-linear non-homogeneous partial differential equations
(PDEs) and derive their exact solution as a functional Taylor expansion in
powers of the source term. The kind of PDEs we consider are dispersive ones
where the exact solution of the corresponding homogeneous equations can have
some known shape. The technique has a formal similarity with the
Dyson--Schwinger set of equations to solve quantum field theories. However,
there are no physical constraints. Indeed, we show that a complete coincidence
with the statistical field model of a quartic scalar theory can be achieved in
the Gaussian expansion of the cumulants of the partition function. | math-ph |
Spectral gaps of Dirac operators describing graphene quantum dots: The two-dimensional Dirac operator describes low-energy excitations in
graphene. Different choices for the boundary conditions give rise to
qualitative differences in the spectrum of the resulting operator. For a family
of boundary conditions, we find a lower bound to the spectral gap around zero,
proportional to $|\Omega|^{-1/2}$, where $\Omega \subset \mathbb{R}^2$ is the
bounded region where the Dirac operator acts. This family contains the
so-called infinite mass and armchair cases used in the physics literature for
the description of graphene quantum dots. | math-ph |
On Thermodynamic and Ultraviolet Stability of Bosonic Lattice QCD Models
in Euclidean Spacetime Dimensions $d=2,3,4$: We prove stability bounds for local gauge-invariant scalar QCD quantum
models, with multiflavored bosons replacing (anti)quarks. We take a compact,
connected gauge Lie group G, and concentrate on G=U(N),SU(N). Let
d(N)=N^2,(N^2-1) be their Lie algebra dimensions. We start on a finite
hypercubic lattice \Lambda\subset aZ^d, d=2,3,4, a\in(0,1], with L sites on a
side, \Lambda_s=L^d sites, and free boundary conditions. The action is a sum of
a Bose-gauge part and a Wilson pure-gauge plaquette term. We employ a priori
local, scaled scalar bosons with an a-dependent field-strength renormalization:
a non-canonical scaling. The Wilson action is a sum over pointwise positive
plaquette actions with a pre-factor (a^{d-4}/g^2), and gauge coupling
$0<g^2\leq g_0^2<\infty$. Sometimes we use an enhanced temporal gauge. Here,
there are \Lambda_r\simeq (d-1)\Lambda_s retained bond variables. The unscaled
partition function is $Z^u_{\Lambda,a}\equiv
Z^u_{\Lambda,a,\kappa_u^2,m_u,g^2,d}$, where $\kappa_u^2>0$ is the unscaled
hopping parameter and m_u are the boson bare masses. Letting $s_B\equiv
[a^{d-2}(m_u^2a^2+2d\kappa_u^2)]^{1/2}$, $s_Y\equiv a^{(d-4)/2}/g$, we show
that the scaled partition function
$Z_{\Lambda,a}=s_B^{N\Lambda_s}s_Y^{d(N)\Lambda_r} Z^u_{\Lambda,a}$ satisfies
the stability bounds $e^{c_\ell d(N)\Lambda_s}\leq Z_{\Lambda,a}\leq
e^{c_ud(N)\Lambda_s}$ with finite real $c_\ell, c_u$ independent of $L$ and the
spacing $a$. We have extracted in $Z^u_{\Lambda,a}$ the dependence on \Lambda
and the exact singular behavior of the finite lattice free energy in the
continuum limit $a\searrow 0$. For the normalized finite-lattice free energy
$f_\Lambda^n=[d(N)\Lambda_s]^{-1}\ln Z_{\Lambda,a}$, we prove the existence of
(at least, subsequentials) a thermodynamic limit for f_\Lambda^n and, next, of
a continuum limit. | math-ph |
Construction of Doubly Periodic Solutions via the Poincare-Lindstedt
Method in the case of Massless Phi^4 Theory: Doubly periodic (periodic both in time and in space) solutions for the
Lagrange-Euler equation of the (1+1)-dimensional scalar Phi^4 theory are
considered. The nonlinear term is assumed to be small, and the
Poincare-Lindstedt method is used to find asymptotic solutions in the standing
wave form. The principal resonance problem, which arises for zero mass, is
solved if the leading-order term is taken in the form of a Jacobi elliptic
function. It have been proved that the choice of elliptic cosine with fixed
value of module k (k=0.451075598811) as the leading-order term puts the
principal resonance to zero and allows us constructed (with accuracy to third
order of small parameter) the asymptotic solution in the standing wave form. To
obtain this leading-order term the computer algebra system REDUCE have been
used. We have appended the REDUCE program to this paper. | math-ph |
The propagator of the attractive delta-Bose gas in one dimension: We consider the quantum delta-Bose gas on the infinite line. For repulsive
interactions, Tracy and Widom have obtained an exact formula for the quantum
propagator. In our contribution we explicitly perform its analytic continuation
to attractive interactions. We also study the connection to the expansion of
the propagator in terms of the Bethe ansatz eigenfunctions. Thereby we provide
an independent proof of their completeness. | math-ph |
Localization for the Ising model in a transverse field with generic
aperiodic disorder: We show that the transverse field Ising model undergoes a zero temperature
phase transition for a $G_\delta$ set of ergodic transverse fields. We apply
our results to the special case of quasiperiodic transverse fields, in one
dimension we find a sharp condition for the existence of a phase transition. | math-ph |
A strong operator topology adiabatic theorem: We prove an adiabatic theorem for the evolution of spectral data under a weak
additive perturbation in the context of a system without an intrinsic time
scale. For continuous functions of the unperturbed Hamiltonian the convergence
is in norm while for a larger class functions, including the spectral
projections associated to embedded eigenvalues, the convergence is in the
strong operator topology. | math-ph |
A rigorous model reduction for the anisotropic-scattering transport
process: In this letter, we propose a reduced-order model to bridge the particle
transport mechanics and the macroscopic fluid dynamics in the highly scattered
regime. A rigorous mathematical derivation and a concise physical
interpretation are presented for an anisotropic-scattering transport process
with arbitrary order of scattering kernel. The prediction of the theoretical
model perfectly agrees with the numerical experiments. A clear picture of the
diffusion physics is revealed for the neutral particle transport in the
asymptotic optically thick regime. | math-ph |
Spectral curve duality beyond the two-matrix model: We describe a simple algebraic approach to several spectral duality results
for integrable systems and illustrate the method for two types of examples: The
Bertola-Eynard-Harnad spectral duality of the two-matrix model as well as the
various dual descriptions of minimal model conformal field theories coupled to
gravity. | math-ph |
Decay estimates for steady solutions of the Navier-Stokes equations in
two dimensions in the presence of a wall: Let w be the vorticity of a stationary solution of the two-dimensional
Navier-Stokes equations with a drift term parallel to the boundary in the
half-plane -\infty<x<\infty, y>1, with zero Dirichlet boundary conditions at
y=1 and at infinity, and with a small force term of compact support. Then,
|xyw(x,y)| is uniformly bounded in the half-plane. The proof is given in a
specially adapted functional framework and complements previous work. | math-ph |
General properties of the Foldy-Wouthuysen transformation and
applicability of the corrected original Foldy-Wouthuysen method: General properties of the Foldy-Wouthuysen transformation which is widely
used in quantum mechanics and quantum chemistry are considered. Merits and
demerits of the original Foldy-Wouthuysen transformation method are analyzed.
While this method does not satisfy the Eriksen condition of the
Foldy-Wouthuysen transformation, it can be corrected with the use of the
Baker-Campbell-Hausdorff formula. We show a possibility of such a correction
and propose an appropriate algorithm of calculations. An applicability of the
corrected Foldy-Wouthuysen method is restricted by the condition of convergence
of a series of relativistic corrections. | math-ph |
A remark on the attainable set of the Schrödinger equation: We discuss the set of wavefunctions $\psi_V(t)$ that can be obtained from a
given initial condition $\psi_0$ by applying the flow of the Schr\"odinger
operator $-\Delta + V(t,x)$ and varying the potential $V(t,x)$. We show that
this set has empty interior, both as a subset of the sphere in
$L^2(\mathbb{R}^d)$ and as a set of trajectories. | math-ph |
On Generalized Diffusion and Heat Systems on an Evolving Surface with a
Boundary: We consider a diffusion process on an evolving surface with a piecewise
Lipschitz-continuous boundary from an energetic point of view. We employ an
energetic variational approach with both surface divergence and transport
theorems to derive the generalized diffusion and heat systems on the evolving
surface. Moreover, we investigate the boundary conditions for the two systems
to study the conservation and energy laws of them. As an application, we make a
mathematical model for a diffusion process on an evolving double bubble.
Especially, this paper is devoted to deriving the representation formula for
the unit outer co-normal vector to the boundary of a surface. | math-ph |
Quantum Hellinger distances revisited: This short note aims to study quantum Hellinger distances investigated
recently by Bhatia et al. [Lett. Math. Phys. 109 (2019), 1777-1804] with a
particular emphasis on barycenters. We introduce the family of generalized
quantum Hellinger divergences, that are of the form $\phi(A,B)=\mathrm{Tr}
\left((1-c)A + c B - A \sigma B \right),$ where $\sigma$ is an arbitrary
Kubo-Ando mean, and $c \in (0,1)$ is the weight of $\sigma.$ We note that these
divergences belong to the family of maximal quantum $f$-divergences, and hence
are jointly convex and satisfy the data processing inequality (DPI). We derive
a characterization of the barycenter of finitely many positive definite
operators for these generalized quantum Hellinger divergences. We note that the
characterization of the barycenter as the weighted multivariate $1/2$-power
mean, that was claimed in the work of Bhatia et al. mentioned above, is true in
the case of commuting operators, but it is not correct in the general case. | math-ph |
Some inequalities for quantum Tsallis entropy related to the strong
subadditivity: In this paper we investigate the inequality $S_q(\rho_{123})+S_q(\rho_2)\leq
S_q(\rho_{12})+S_q(\rho_{23}) \, (*)$ where $\rho_{123}$ is a state on a finite
dimensional Hilbert space $\mathcal{H}_1\otimes \mathcal{H}_2\otimes
\mathcal{H}_3,$ and $S_q$ is the Tsallis entropy. It is well-known that the
strong subadditivity of the von Neumnann entropy can be derived from the
monotonicity of the Umegaki relative entropy. Now, we present an equivalent
form of $(*)$, which is an inequality of relative quasi-entropies. We derive an
inequality of the form $S_q(\rho_{123})+S_q(\rho_2)\leq
S_q(\rho_{12})+S_q(\rho_{23})+f_q(\rho_{123})$, where $f_1(\rho_{123})=0$. Such
a result can be considered as a generalization of the strong subadditivity of
the von Neumnann entropy. One can see that $(*)$ does not hold in general (a
picturesque example is included in this paper), but we give a sufficient
condition for this inequality, as well. | math-ph |
PT-Invariant Periodic Potentials with a Finite Number of Band Gaps: We obtain the band edge eigenstates and the mid-band states for the complex,
PT-invariant generalized associated Lam\'e potentials $V^{PT}(x)=-a(a+1)m
\sn^2(y,m)-b(b+1)m {\sn^2 (y+K(m),m)} -f(f+1)m {\sn^2
(y+K(m)+iK'(m),m)}-g(g+1)m {\sn^2 (y+iK'(m),m)}$, where $y \equiv ix+\beta$,
and there are four parameters $a,b,f,g$. This work is a substantial
generalization of previous work with the associated Lam\'e potentials
$V(x)=a(a+1)m\sn^2(x,m)+b(b+1)m{\sn^2 (x+K(m),m)}$ and their corresponding
PT-invariant counterparts $V^{PT}(x)=-V(ix+\beta)$, both of which involving
just two parameters $a,b$. We show that for many integer values of $a,b,f,g$,
the PT-invariant potentials $V^{PT}(x)$ are periodic problems with a finite
number of band gaps. Further, usingsupersymmetry, we construct several
additional, new, complex, PT-invariant, periodic potentials with a finite
number of band gaps. We also point out the intimate connection between the
above generalized associated Lam\'e potential problem and Heun's differential
equation. | math-ph |
Topological recursion on the Bessel curve: The Witten-Kontsevich theorem states that a certain generating function for
intersection numbers on the moduli space of stable curves is a tau-function for
the KdV integrable hierarchy. This generating function can be recovered via the
topological recursion applied to the Airy curve $x=\frac{1}{2}y^2$. In this
paper, we consider the topological recursion applied to the irregular spectral
curve $xy^2=\frac{1}{2}$, which we call the Bessel curve. We prove that the
associated partition function is also a KdV tau-function, which satisfies
Virasoro constraints, a cut-and-join type recursion, and a quantum curve
equation. Together, the Airy and Bessel curves govern the local behaviour of
all spectral curves with simple branch points. | math-ph |
Dynamics of a planar Coulomb gas: We study the long-time behavior of the dynamics of interacting planar
Brow-nian particles, confined by an external field and subject to a singular
pair repulsion. The invariant law is an exchangeable Boltzmann -- Gibbs
measure. For a special inverse temperature, it matches the Coulomb gas known as
the complex Ginibre ensemble. The difficulty comes from the interaction which
is not convex, in contrast with the case of one-dimensional log-gases
associated with the Dyson Brownian Motion. Despite the fact that the invariant
law is neither product nor log-concave, we show that the system is well-posed
for any inverse temperature and that Poincar{\'e} inequalities are available.
Moreover the second moment dynamics turns out to be a nice Cox -- Ingersoll --
Ross process in which the dependency over the number of particles leads to
identify two natural regimes related to the behavior of the noise and the speed
of the dynamics. | math-ph |
QKZ-Ruijsenaars correspondence revisited: We discuss the Matsuo-Cherednik type correspondence between the quantum
Knizhnik-Zamolodchikov equations associated with $GL(N)$ and the $n$-particle
quantum Ruijsenaars model, with $n$ being not necessarily equal to $N$. The
quasiclassical limit of this construction yields the quantum-classical
correspondence between the quantum spin chains and the classical Ruijsenaars
models. | math-ph |
Quantum macrostatistical picture of nonequilibrium steady states: We employ a quantum macrostatistical treatment of irreversible processes to
prove that, in nonequilibrium steady states, (a) the hydrodynamical observables
execute a generalised Onsager-Machlup process and (b) the spatial correlations
of these observables are generically of long range. The key assumptions behind
these results are a nonequilibrium version of Onsager's regression hypothesis,
together with certain hypotheses of chaoticity and local equilibrium for
hydrodynamical fluctuations. | math-ph |
Speedy motions of a body immersed in an infinitely extended medium: We study the motion of a classical point body of mass M, moving under the
action of a constant force of intensity E and immersed in a Vlasov fluid of
free particles, interacting with the body via a bounded short range potential
Psi. We prove that if its initial velocity is large enough then the body
escapes to infinity increasing its speed without any bound "runaway effect".
Moreover, the body asymptotically reaches a uniformly accelerated motion with
acceleration E/M. We then discuss at a heuristic level the case in which Psi(r)
diverges at short distances like g r^{-a}, g,a>0, by showing that the runaway
effect still occurs if a<2. | math-ph |
Surface Energies Arising in Microscopic Modeling of Martensitic
Transformations: In this paper we construct and analyze a two-well Hamiltonian on a 2D atomic
lattice. The two wells of the Hamiltonian are prescribed by two rank-one
connected martensitic twins, respectively. By constraining the deformed
configurations to special 1D atomic chains with position-dependent elongation
vectors for the vertical direction, we show that the structure of ground states
under appropriate boundary conditions is close to the macroscopically expected
twinned configurations with additional boundary layers localized near the
twinning interfaces. In addition, we proceed to a continuum limit, show
asymptotic piecewise rigidity of minimizing sequences and rigorously derive the
corresponding limiting form of the surface energy. | math-ph |
Diffusion limit for a kinetic equation with a thermostatted interface: We consider a linear phonon Boltzmann equation with a
reflecting/transmitting/absorbing interface. This equation appears as the
Boltzmann-Grad limit for the energy density function of a harmonic chain of
oscillators with inter-particle stochastic scattering in the presence of a heat
bath at temperature $T$ in contact with one oscillator at the origin. We prove
that under the diffusive scaling the solutions of the phonon equation tend to
the solution $\rho(t,y)$ of a heat equation with the boundary condition
$\rho(t,0)\equiv T$. | math-ph |
Generalized Scallop Theorem for Linear Swimmers: In this article, we are interested in studying locomotion strategies for a
class of shape-changing bodies swimming in a fluid. This class consists of
swimmers subject to a particular linear dynamics, which includes the two most
investigated limit models in the literature: swimmers at low and high Reynolds
numbers. Our first contribution is to prove that although for these two models
the locomotion is based on very different physical principles, their dynamics
are similar under symmetry assumptions. Our second contribution is to derive
for such swimmers a purely geometric criterion allowing to determine wether a
given sequence of shape-changes can result in locomotion. This criterion can be
seen as a generalization of Purcell's scallop theorem (stated in Purcell
(1977)) in the sense that it deals with a larger class of swimmers and address
the complete locomotion strategy, extending the usual formulation in which only
periodic strokes for low Reynolds swimmers are considered. | math-ph |
On complex structures in physics: Complex numbers enter fundamental physics in at least two rather distinct
ways. They are needed in quantum theories to make linear differential operators
into Hermitian observables. Complex structures appear also, through Hodge
duality, in vector and spinor spaces associated with space-time. This paper
reviews some of these notions. Charge conjugation in multidimensional
geometries and the appearance of Cauchy-Riemann structures on Lorentz manifolds
with a congruence of null geodesics without shear are presented in considerable
detail. | math-ph |
Evaluation of the second virial coefficient for the Mie potential using
the method of brackets: The second virial coefficient for the Mie potential is evaluated using the
method of brackets. This method converts a definite integral into a series in
the parameters of the problem, in this case this is the temperature $T$. The
results obtained here are consistent with some known special cases, such as the
Lenard-Jones potential. The asymptotic properties of the second virial
coefficient in molecular thermodynamic systems and complex fluid modeling are
described in the limiting cases of $T \rightarrow 0$ and $T \rightarrow
\infty$. | math-ph |
Microscopic solutions of the Boltzmann-Enskog equation in the series
representation: The Boltzmann-Enskog equation for a hard sphere gas is known to have so
called microscopic solutions, i.e., solutions of the form of time-evolving
empirical measures of a finite number of hard spheres. However, the precise
mathematical meaning of these solutions should be discussed, since the formal
substitution of empirical measures into the equation is not well-defined. Here
we give a rigorous mathematical meaning to the microscopic solutions to the
Boltzmann-Enskog equation by means of a suitable series representation. | math-ph |
Asymptotic morphisms and superselection theory in the scaling limit: Given a local Haag-Kastler net of von Neumann algebras and one of its scaling
limit states, we introduce a variant of the notion of asymptotic morphism by
Connes and Higson, and we show that the unitary equivalence classes of
(localized) morphisms of the scaling limit theory of the original net are in
bijection with classes of suitable pairs of such asymptotic morphisms. In the
process, we also show that the quasi-local C*-algebras of two nets are
isomorphic under very general hypotheses, and we construct an extension of the
scaling algebra whose representation on the scaling limit Hilbert space
contains the local von Neumann algebras. We also study the relation between our
asymptotic morphisms and superselection sectors preserved in the scaling limit. | math-ph |
Averages over Ginibre's Ensemble of Random Real Matrices: We give a method for computing the ensemble average of multiplicative class
functions over the Gaussian ensemble of real asymmetric matrices. These
averages are expressed in terms of the Pfaffian of Gram-like antisymmetric
matrices formed with respect to a skew-symmetric inner product related to the
class function. | math-ph |
Z-measures on partitions and their scaling limits: We study certain probability measures on partitions of n=1,2,..., originated
in representation theory, and demonstrate their connections with random matrix
theory and multivariate hypergeometric functions.
Our measures depend on three parameters including an analog of the beta
parameter in random matrix models. Under an appropriate limit transition as n
goes to infinity, our measures converge to certain limit measures, which are of
the same nature as one-dimensional log-gas with arbitrary beta>0.
The first main result says that averages of products of ``characteristic
polynomials'' with respect to the limit measures are given by the multivariate
hypergeometric functions of type (2,0). The second main result is a computation
of the limit correlation functions for the even values of beta. | math-ph |
Entanglement for multipartite systems of indistinguishable particles: We analyze the concept of entanglement for multipartite system with bosonic
and fermionic constituents and its generalization to systems with arbitrary
parastatistics. We use the representation theory of symmetry groups to
formulate a unified approach to this problem in terms of simple tensors with
appropriate symmetry. For an arbitrary parastatistics, we define the S-rank
generalizing the notion of the Schmidt rank. The S-rank, defined for all types
of tensors, serves for distinguishing entanglement of pure states. In addition,
for Bose and Fermi statistics, we construct an analog of the Jamiolkowski
isomorphism. | math-ph |
Infinite energy solutions to inelastic homogeneous Boltzmann equation: This paper is concerned with the existence, shape and dynamical stability of
infinite-energy equilibria for a general class of spatially homogeneous kinetic
equations in space dimensions $d \geq 3$. Our results cover in particular
Bobyl\"ev's model for inelastic Maxwell molecules. First, we show under certain
conditions on the collision kernel, that there exists an index $\alpha\in(0,2)$
such that the equation possesses a nontrivial stationary solution, which is a
scale mixture of radially symmetric $\alpha$-stable laws. We also characterize
the mixing distribution as the fixed point of a smoothing transformation.
Second, we prove that any transient solution that emerges from the NDA of some
(not necessarily radial symmetric) $\alpha$-stable distribution converges to an
equilibrium. The key element of the convergence proof is an application of the
central limit theorem to a representation of the transient solution as a
weighted sum of i.i.d. random vectors. | math-ph |
Geometric Phase and Modulo Relations for Probability Amplitudes as
Functions on Complex Parameter Spaces: We investigate general differential relations connecting the respective
behavior s of the phase and modulo of probability amplitudes of the form
$\amp{\psi_f}{\psi}$, where $\ket{\psi_f}$ is a fixed state in Hilbert space
and $\ket{\psi}$ is a section of a holomorphic line bundle over some complex
parameter space. Amplitude functions on such bundles, while not strictly
holomorphic, nevertheless satisfy generalized Cauchy-Riemann conditions
involving the U(1) Berry-Simon connection on the parameter space. These
conditions entail invertible relations between the gradients of the phase and
modulo, therefore allowing for the reconstruction of the phase from the modulo
(or vice-versa) and other conditions on the behavior of either polar component
of the amplitude. As a special case, we consider amplitude functions valued on
the space of pure states, the ray space ${\cal R} = {\mathbb C}P^n$, where
transition probabilities have a geometric interpretation in terms of geodesic
distances as measured with the Fubini-Study metric. In conjunction with the
generalized Cauchy-Riemann conditions, this geodesic interpretation leads to
additional relations, in particular a novel connection between the modulus of
the amplitude and the phase gradient, somewhat reminiscent of the WKB formula.
Finally, a connection with geometric phases is established. | math-ph |
Differential Geometry on SU(3) with Applications to Three State Systems: The left and right invariant vector fields are calculated in an ``Euler
angle'' type parameterization for the group manifold of SU(3), referred to here
as Euler coordinates. The corresponding left and right invariant one-forms are
then calculated. This enables the calculation of the invariant volume element
or Haar measure. These are then used to describe the density matrix of a pure
state and geometric phases for three state systems. | math-ph |
Epsilon-complexity of continuous functions: A formal definition of epsilon-complexity of an individual continuous
function defined on a unit cube is proposed. This definition is consistent with
the Kolmogorov's idea of the complexity of an object. A definition of
epsilon-complexity for a class of continuous functions with a given modulus of
continuity is also proposed. Additionally, an explicit formula for the
epsilon-complexity of a functional class is obtained. As a consequence, the
paper finds that the epsilon-complexity for the Holder class of functions can
be characterized by a pair of real numbers. Based on these results the papers
formulates a conjecture concerning the epsilon-complexity of an individual
function from the Holder class. We also propose a conjecture about
characterization of epsilon-complexity of a function from the Holder class
given on a discrete grid. | math-ph |
Conformal and Contact Kinetic Dynamics and Their Geometrization: We propose a conformal generalization of the reversible Vlasov equation of
kinetic plasma dynamics, called conformal kinetic theory. In order to arrive at
this formalism, we start with the conformal Hamiltonian dynamics of particles
and lift it to the dynamical formulation of the associated kinetic theory. The
resulting theory represents a simple example of a geometric pathway from
dissipative particle motion to dissipative kinetic motion. We also derive the
kinetic equations of a continuum of particles governed by the contact
Hamiltonian dynamics, which may be interpreted in the context of relativistic
mechanics. Once again we start with the contact Hamiltonian dynamics and lift
it to a kinetic theory, called contact kinetic dynamics. Finally, we project
the contact kinetic theory to conformal kinetic theory so that they form a
geometric hierarchy. | math-ph |
Arctic curves of the four-vertex model: We consider the four-vertex model with a special choice of fixed boundary
conditions giving rise to limit shape phenomena. More generally, the considered
boundary conditions relate vertex models to scalar products of off-shell Bethe
states, boxed plane partitions, and fishnet diagrams in quantum field theory.
In the scaling limit, the model exhibits the emergence of an arctic curve
separating a central disordered region from six frozen `corners' of
ferroelectric or anti-ferroelectric type. We determine the analytic expression
of the interface by means of the Tangent Method. We supplement this heuristic
method with an alternative, rigorous derivation of the arctic curve. This is
based on the exact evaluation of suitable correlation functions, devised to
detect spatial transition from order to disorder, in terms of the partition
function of some discrete log-gas associated to the orthogonalizing measure of
the Hahn polynomials. As a by-product, we also deduce that the arctic curve's
fluctuations are governed by the Tracy-Widom distribution. | math-ph |
Generalized MICZ-Kepler system, duality, polynomial and deformed
oscillator algebras: We present the quadratic algebra of the generalized MICZ-Kepler system in
three-dimensional Euclidean space $E_{3}$ and its dual the four dimensional
singular oscillator in four-dimensional Euclidean space $E_{4}$. We present
their realization in terms of a deformed oscillator algebra using the
Daskaloyannis construction. The structure constants are in these cases function
not only of the Hamiltonian but also of other integrals commuting with all
generators of the quadratic algebra. We also present a new algebraic derivation
of the energy spectrum of the MICZ-Kepler system on the three sphere $S^{3}$
using a quadratic algebra. These results point out also that results and
explicit formula for structure functions obtained for quadratic, cubic and
higher order polynomial algebras in context of two-dimensional superintegrable
systems may be applied to superintegrable systems in higher dimensions with and
without monopoles. | math-ph |
Noninertial effects on a Dirac neutral particle inducing an analogue of
the Landau quantization in the cosmic string spacetime: We discuss the behaviour of external fields that interact with a Dirac
neutral particle with a permanent electric dipole moment in order to achieve
relativistic bound states solutions in a noninertial frame and in the presence
of a topological defect spacetime. We show that the noninertial effects of the
Fermi-Walker reference frame induce a radial magnetic field even in the absence
of magnetic charges, which is influenced by the topology of the cosmic string
spacetime. We then discuss the conditions that the induced fields must satisfy
to yield the relativistic bound states corresponding to the
Landau-He-McKellar-Wilkens quantization in the cosmic string spacetime. Finally
we obtain the Dirac spinors for positive-energy solutions and the Gordon
decomposition of the Dirac probability current. | math-ph |
On Random Matrix Averages Involving Half-Integer Powers of GOE
Characteristic Polynomials: Correlation functions involving products and ratios of half-integer powers of
characteristic polynomials of random matrices from the Gaussian Orthogonal
Ensemble (GOE) frequently arise in applications of Random Matrix Theory (RMT)
to physics of quantum chaotic systems, and beyond. We provide an explicit
evaluation of the large-$N$ limits of a few non-trivial objects of that sort
within a variant of the supersymmetry formalism, and via a related but
different method. As one of the applications we derive the distribution of an
off-diagonal entry $K_{ab}$ of the resolvent (or Wigner $K$-matrix) of GOE
matrices which, among other things, is of relevance for experiments on chaotic
wave scattering in electromagnetic resonators. | math-ph |
The tunneling hamiltonian representation of false vaccuum decay: II.
Application to soliton - anti soliton pair creation: The tunneling hamiltonian has proven to be a useful method in many body
physics to treat particle tunneling between different states represented as
wavefunctions. Our problem is here applying what we did in the first paper to a
driven sine-Gordon system. Here we apply a generalization of the tunneling
Hamiltonian to charge density wave transport problems, in which tunneling
between states which are wavefunctionals of a scalar quantum field are
considered. We derive I-E curves which match Zenier curves used to fit data
experimentally with wavefunctionals congruent with the false vacuum hypothesis | math-ph |
Barrier methods for critical exponent problems in geometric analysis and
mathematical physics: We consider the design and analysis of numerical methods for approximating
positive solutions to nonlinear geometric elliptic partial differential
equations containing critical exponents. This class of problems includes the
Yamabe problem and the Einstein constraint equations, which simultaneously
contain several challenging features: high spatial dimension n >= 3, varying
(potentially non-smooth) coefficients, critical (even super-critical)
nonlinearity, non-monotone nonlinearity (arising from a non-convex energy), and
spatial domains that are typically Riemannian manifolds rather than simply open
sets in Rn. These problems may exhibit multiple solutions, although only
positive solutions typically have meaning. This creates additional complexities
in both the theory and numerical treatment of such problems, as this feature
introduces both non-uniqueness as well as the need to incorporate an inequality
constraint into the formulation. In this work, we consider numerical methods
based on Galerkin-type discretization, covering any standard bases construction
(finite element, spectral, or wavelet), and the combination of a barrier method
for nonconvex optimization and global inexact Newton-type methods for dealing
with nonconvexity and the presence of inequality constraints. We first give an
overview of barrier methods in non-convex optimization, and then develop and
analyze both a primal barrier energy method for this class of problems. We then
consider a sequence of numerical experiments using this type of barrier method,
based on a particular Galerkin method, namely the piecewise linear finite
element method, leverage the FETK modeling package. We illustrate the behavior
of the primal barrier energy method for several examples, including the Yamabe
problem and the Hamiltonian constraint. | math-ph |
On the nature of the Tsallis-Fourier Transform: By recourse to tempered ultradistributions, we show here that the effect of a
q-Fourier transform (qFT) is to map {\it equivalence classes} of functions into
other classes in a one-to-one fashion. This suggests that Tsallis' q-statistics
may revolve around equivalence classes of distributions and not on individual
ones, as orthodox statistics does. We solve here the qFT's non-invertibility
issue, but discover a problem that remains open. | math-ph |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.