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Products of Rectangular Random Matrices: Singular Values and Progressive
Scattering: We discuss the product of $M$ rectangular random matrices with independent
Gaussian entries, which have several applications including wireless
telecommunication and econophysics. For complex matrices an explicit expression
for the joint probability density function is obtained using the
Harish-Chandra--Itzykson--Zuber integration formula. Explicit expressions for
all correlation functions and moments for finite matrix sizes are obtained
using a two-matrix model and the method of bi-orthogonal polynomials. This
generalises the classical result for the so-called Wishart--Laguerre Gaussian
unitary ensemble (or chiral unitary ensemble) at M=1, and previous results for
the product of square matrices. The correlation functions are given by a
determinantal point process, where the kernel can be expressed in terms of
Meijer $G$-functions. We compare the results with numerical simulations and
known results for the macroscopic level density in the limit of large matrices.
The location of the endpoints of support for the latter are analysed in detail
for general $M$. Finally, we consider the so-called ergodic mutual information,
which gives an upper bound for the spectral efficiency of a MIMO communication
channel with multi-fold scattering. | math-ph |
Emerging Jordan forms, with applications to critical statistical models
and conformal field theory: Two novel frameworks for handling mathematical and physical problems are
introduced. The first, the emerging Jordan form, generalizes the concept of the
Jordan canonical form, a well-established tool of linear algebra. The second,
dual Jordan quantum physics, generalizes the framework of quantum physics to
one in which the hermiticity postulate is considerably relaxed. These
frameworks are then used to resolve some long-outstanding problems in
theoretical physics, coming from critical statistical models and conformal
field theory. I describe these problems and the difficulties involved in
finding satisfactory solutions, then show how the concepts of emerging Jordan
forms and dual Jordan quantum physics are naturally suited to overcoming these
difficulties. Although their applications in this work are limited in scope to
rather specific problems, the frameworks themselves are completely general, and
I describe ways in which they may be used in other areas of mathematics and
physics. Several appendices close the work, which include improvements to a
widely used computational algorithm and corrections to some published data. | math-ph |
Dislocation Defects and Diophantine Approximation: In this paper we consider a Schrodinger eigenvalue problem with a potential
consisting of a periodic part together with a compactly supported defect
potential. Such problems arise as models in condensed matter to describe color
in crystals as well as in engineering to describe optical photonic structures.
We are interested in studying the existence of point eigenvalues in gaps in the
essential spectrum, and in particular in counting the number of such
eigenvalues. We use a homotopy argument in the width of the potential to count
the eigenvalues as they are created. As a consequence of this we prove the
following significant generalization of Zheludev's theorem: the number of point
eigenvalues in a gap in the essential spectrum is exactly one for sufficiently
large gap number unless a certain Diophantine approximation problem has
solutions, in which case there exists a subsequence of gaps containing 0,1 or 2
eigenvalues. We state some conditions under which the solvability of the
Diophantine approximation problem can be established. | math-ph |
Phase transition between two-component and three-component ground states
of spin-1 Bose-Einstein condensates: For an antiferromagnetic spin-1 Bose-Einstein condensate under an applied
uniform magnetic field, its ground state $(\psi_1,\psi_0,\psi_{-1})$ undergoes
a phase transition from a two-component state ($\psi_0 \equiv 0$) to a
three-component state ($\psi_j\ne 0$ for all $j$) at a critical value of the
magnetic field. This phenomenon has been observed in numerical simulations as
well as in experiments. In this paper, we provide a mathematical proof based on
a simple principle found by the authors: a redistribution of the mass densities
between different components will decrease the kinetic energy. | math-ph |
Graphical functions in parametric space: Graphical functions are positive functions on the punctured complex plane
$\mathbb{C}\setminus\{0,1\}$ which arise in quantum field theory. We generalize
a parametric integral representation for graphical functions due to Lam, Lebrun
and Nakanishi, which implies the real analyticity of graphical functions.
Moreover we prove a formula that relates graphical functions of planar dual
graphs. | math-ph |
Jacobi Ensemble, Hurwitz Numbers and Wilson Polynomials: We express the topological expansion of the Jacobi Unitary Ensemble in terms
of triple monotone Hurwitz numbers. This completes the combinatorial
interpretation of the topological expansion of the classical unitary invariant
matrix ensembles. We also provide effective formulae for generating functions
of multipoint correlators of the Jacobi Unitary Ensemble in terms of Wilson
polynomials, generalizing the known relations between one point correlators and
Wilson polynomials. | math-ph |
Superposition rules for higher-order systems and their applications: Superposition rules form a class of functions that describe general solutions
of systems of first-order ordinary differential equations in terms of generic
families of particular solutions and certain constants. In this work we extend
this notion and other related ones to systems of higher-order differential
equations and analyse their properties. Several results concerning the
existence of various types of superposition rules for higher-order systems are
proved and illustrated with examples extracted from the physics and mathematics
literature. In particular, two new superposition rules for second- and
third-order Kummer--Schwarz equations are derived. | math-ph |
Nambu brackets for the electromagnetic field: A Nambu formulation for the electromagnetic field in the case of stationary
charge density and vanishing charge current density is proposed. | math-ph |
Level sets percolation on chaotic graphs: One of the most surprising discoveries in quantum chaos was that nodal
domains of eigenfunctions of quantum-chaotic billiards and maps in the
semi-classical limit display critical percolation. Here we extend these studies
to the level sets of the adjacency eigenvectors of d-regular graphs. Numerical
computations show that the statistics of the largest level sets (the maximal
connected components of the graph for which the eigenvector exceeds a
prescribed value) depend critically on the level. The critical level is a
function of the eigenvalue and the degree d. To explain the observed behavior
we study a random Gaussian waves ensemble over the d-regular tree. For this
model, we prove the existence of a critical threshold. Using the local tree
property of d-regular graphs, and assuming the (local) applicability of the
random waves model, we can compute the critical percolation level and reproduce
the numerical simulations. These results support the random-waves model for
random regular graphs and provides an extension to Bogomolny's percolation
model for two-dimensional chaotic billiards. | math-ph |
Motion by curvature and large deviations for an interface dynamics on Z
2: We study large deviations for a Markov process on curves in Z 2 mimicking the
motion of an interface. Our dynamics can be tuned with a parameter $\beta$,
which plays the role of an inverse temperature, and coincides at $\beta$ =
$\infty$ with the zero-temperature Ising model with Glauber dynamics, where
curves correspond to the boundaries of droplets of one phase immersed in a sea
of the other one. We prove that contours typically follow a motion by curvature
with an influence of the parameter $\beta$, and establish large deviations
bounds at all large enough $\beta$ < $\infty$. The diffusion coefficient and
mobility of the model are identified and correspond to those predicted in the
literature. | math-ph |
Fusion for the one-dimensional Hubbard model: We discuss a formulation of the fusion procedure for integrable models which
is suitable for application to non-standard R-matrices. It allows for
construction of bound state R-matrices for AdS/CFT worldsheet scattering or
equivalently for the one-dimensional Hubbard model. We also discuss some
peculiar cases that arise in these models. | math-ph |
Mean hitting times of quantum Markov chains in terms of generalized
inverses: We study quantum Markov chains on graphs, described by completely positive
maps, following the model due to S. Gudder (J. Math. Phys. 49, 072105, 2008)
and which includes the dynamics given by open quantum random walks as defined
by S. Attal et al. (J. Stat. Phys. 147:832-852, 2012). After reviewing such
structures we examine a quantum notion of mean time of first visit to a chosen
vertex. However, instead of making direct use of the definition as it is
usually done, we focus on expressions for such quantity in terms of generalized
inverses associated with the walk and most particularly the so-called
fundamental matrix. Such objects are in close analogy with the theory of Markov
chains and the methods described here allow us to calculate examples that
illustrate similarities and differences between the quantum and classical
settings. | math-ph |
Spectral analysis of finite-time correlation matrices near equilibrium
phase transitions: We study spectral densities for systems on lattices, which, at a phase
transition display, power-law spatial correlations. Constructing the spatial
correlation matrix we prove that its eigenvalue density shows a power law that
can be derived from the spatial correlations. In practice time series are short
in the sense that they are either not stationary over long time intervals or
not available over long time intervals. Also we usually do not have time series
for all variables available. We shall make numerical simulations on a
two-dimensional Ising model with the usual Metropolis algorithm as time
evolution. Using all spins on a grid with periodic boundary conditions we find
a power law, that is, for large grids, compatible with the analytic result. We
still find a power law even if we choose a fairly small subset of grid points
at random. The exponents of the power laws will be smaller under such
circumstances. For very short time series leading to singular correlation
matrices we use a recently developed technique to lift the degeneracy at zero
in the spectrum and find a significant signature of critical behavior even in
this case as compared to high temperature results which tend to those of random
matrix models. | math-ph |
Generalized Christoffel-Darboux formula for skew-orthogonal polynomials
and random matrix theory: We obtain generalized Christoffel-Darboux (GCD) formula for skew-orthogonal
polynomials (SOP). Using this, we present an alternative derivation of the
level density and two-point function for Gaussian orthogonal ensembles (GOE)
and Gaussian symplectic ensembles (GSE) of random matrices. | math-ph |
Chaotic vibrations and strong scars: This article aims at popularizing some aspects of "quantum chaos", in
particular the study of eigenmodes of classically chaotic systems, in the
semiclassical (or high frequency) limit. | math-ph |
Leibniz algebroid associated with a Nambu-Poisson structure: The notion of Leibniz algebroid is introduced, and it is shown that each
Nambu-Poisson manifold has associated a canonical Leibniz algebroid. This fact
permits to define the modular class of a Nambu-Poisson manifold as an
appropiate cohomology class, extending the well-known modular class of Poisson
manifolds. | math-ph |
Singular perturbation of polynomial potentials in the complex domain
with applications to PT-symmetric families: In the first part of the paper, we discuss eigenvalue problems of the form
-w"+Pw=Ew with complex potential P and zero boundary conditions at infinity on
two rays in the complex plane. We give sufficient conditions for continuity of
the spectrum when the leading coefficient of P tends to 0. In the second part,
we apply these results to the study of topology and geometry of the real
spectral loci of PT-symmetric families with P of degree 3 and 4, and prove
several related results on the location of zeros of their eigenfunctions. | math-ph |
Realization of associative products in terms of Moyal and tomographic
symbols: The quantizer-dequantizer method allows to construct associative products on
any measure space. Here we consider an inverse problem: given an associative
product is it possible to realize it within the quantizer-dequantizer
framework? The answer is positive in finite dimensions and we give a few
examples in infinite dimensions. | math-ph |
Coulomb scattering in the massless Nelson model IV. Atom-electron
scattering: We consider the massless Nelson model with two types of massive particles
which we call atoms and electrons. The atoms interact with photons via an
infrared regular form-factor and thus they are Wigner-type particles with sharp
mass-shells. The electrons have an infrared singular form-factor and thus they
are infraparticles accompanied by soft-photon clouds correlated with their
velocities. In the weak coupling regime we construct scattering states of one
atom and one electron, and demonstrate their asymptotic clustering into
individual particles. The proof relies on the Cook's argument, clustering
estimates, and the non-stationary phase method. The latter technique requires
sharp estimates on derivatives of the ground state wave functions of the fiber
Hamiltonians of the model, which were proven in the earlier papers of this
series. Although we rely on earlier studies of the atom-atom and
electron-photon scattering in the Nelson model, the paper is written in a
self-contained manner. A perspective on the open problem of the
electron-electron scattering in this model is also given. | math-ph |
Propagation of chaos for many-boson systems in one dimension with a
point pair-interaction: We consider the semiclassical limit of nonrelativistic quantum many-boson
systems with delta potential in one dimensional space. We prove that time
evolved coherent states behave semiclassically as squeezed states by a
Bogoliubov time-dependent affine transformation. This allows us to obtain
properties analogous to those proved by Hepp and Ginibre-Velo (\cite{Hep},
\cite{GiVe1,GiVe2}) and also to show propagation of chaos for Schr\"odinger
dynamics in the mean field limit. Thus, we provide a derivation of the cubic
NLS equation in one dimension. | math-ph |
Graded Differential Geometry of Graded Matrix Algebras: We study the graded derivation-based noncommutative differential geometry of
the $Z_2$-graded algebra ${\bf M}(n| m)$ of complex $(n+m)\times(n+m)$-matrices
with the ``usual block matrix grading'' (for $n\neq m$). Beside the
(infinite-dimensional) algebra of graded forms the graded Cartan calculus,
graded symplectic structure, graded vector bundles, graded connections and
curvature are introduced and investigated. In particular we prove the
universality of the graded derivation-based first-order differential calculus
and show, that ${\bf M}(n|m)$ is a ``noncommutative graded manifold'' in a
stricter sense: There is a natural body map and the cohomologies of ${\bf
M}(n|m)$ and its body coincide (as in the case of ordinary graded manifolds). | math-ph |
On Gibbs Measures of Models with Competing Ternary and Binary
Interactions and Corresponding Von Neumann Algebras II: In the present paper the Ising model with competing binary ($J$) and binary
($J_1$) interactions with spin values $\pm 1$, on a Cayley tree of order 2 is
considered. The structure of Gibbs measures for the model considered is
studied. We completely describe the set of all periodic Gibbs measures for the
model with respect to any normal subgroup of finite index of a group
representation of the Cayley tree. Types of von Neumann algebras, generated by
GNS-representation associated with diagonal states corresponding to the
translation invariant Gibbs measures, are determined. It is proved that the
factors associated with minimal and maximal Gibbs states are isomorphic, and if
they are of type III$_\lambda$ then the factor associated with the unordered
phase of the model can be considered as a subfactors of these factors
respectively. Some concrete examples of factors are given too. \\[10mm] {\bf
Keywords:} Cayley tree, Ising model, competing interactions, Gibbs measure,
GNS-construction, Hamiltonian, von Neumann algebra. | math-ph |
Integrable coupled Li$\acute{e}$nard-type systems with balanced loss and
gain: A Hamiltonian formulation of generic many-particle systems with
space-dependent balanced loss and gain coefficients is presented. It is shown
that the balancing of loss and gain necessarily occurs in a pair-wise fashion.
Further, using a suitable choice of co-ordinates, the Hamiltonian can always be
reformulated as a many-particle system in the background of a pseudo-Euclidean
metric and subjected to an analogous inhomogeneous magnetic field with a
functional form that is identical with space-dependent loss/gain
co-efficient.The resulting equations of motion from the Hamiltonian are a
system of coupled Li$\acute{e}$nard-type differential equations. Partially
integrable systems are obtained for two distinct cases, namely, systems with
(i) translational symmetry or (ii) rotational invariance in a pseudo-Euclidean
space. A total number of $m+1$ integrals of motion are constructed for a system
of $2m$ particles, which are in involution, implying that two-particle systems
are completely integrable. A few exact solutions for both the cases are
presented for specific choices of the potential and space-dependent gain/loss
co-efficients, which include periodic stable solutions. Quantization of the
system is discussed with the construction of the integrals of motion for
specific choices of the potential and gain-loss coefficients. A few
quasi-exactly solvable models admitting bound states in appropriate Stoke
wedges are presented. | math-ph |
Integral identities for an interfacial crack in an anisotropic
bimaterial with an imperfect interface: We study a crack lying along an imperfect interface in an anisotropic
bimaterial. A method is devised where known weight functions for the perfect
interface problem are used to obtain singular integral equations relating the
tractions and displacements for both the in-plane and out-of-plane fields. The
integral equations for the out-of-plane problem are solved numerically for
orthotropic bimaterials with differing orientations of anisotropy and for
different extents of interfacial imperfection. These results are then compared
with finite element computations. | math-ph |
Spectral density asymptotics for Gaussian and Laguerre $β$-ensembles
in the exponentially small region: The first two terms in the large $N$ asymptotic expansion of the $\beta$
moment of the characteristic polynomial for the Gaussian and Laguerre
$\beta$-ensembles are calculated. This is used to compute the asymptotic
expansion of the spectral density in these ensembles, in the exponentially
small region outside the leading support, up to terms $o(1)$ . The leading form
of the right tail of the distribution of the largest eigenvalue is given by the
density in this regime. It is demonstrated that there is a scaling from this,
to the right tail asymptotics for the distribution of the largest eigenvalue at
the soft edge. | math-ph |
Recovery of a potential in a fractional diffusion equation: We consider the determination of an unknown potential $q(x)$ form a
fractional diffusion equation subject to overposed lateral boundary data. We
show that this data allows recovery of two spectral sequences for the
associated inverse Sturm-Liouville problem and these are sufficient to apply
standard uniqueness results for this case.
We also look at reconstruction methods and in particular examine the issue of
stability of the solution with respect to the data. The outcome shows the
inverse problem to be severely ill-conditioned and we consider the differences
between the cases of fractional and of classical diffusion. | math-ph |
The Leading Behaviour of The Ground-State Energy of Heavy Ions According
to Brown and Ravenhall: In this article we prove the absence of relativistic effects in leading order
for the ground-state energy according to Brown-Ravenhall operator. We obtain
this asymptotic result for negative ions and for systems with the number of
electrons proportional to the nuclear charge. In the case of neutral atoms the
analogous result was obtained earlier by Cassanas and Siedentop [4]. | math-ph |
Some Algebraic Aspects of the Inhomogeneous Six-Vertex Model: The inhomogeneous six-vertex model is a 2$D$ multiparametric integrable
statistical system. In the scaling limit it is expected to cover different
classes of critical behaviour which, for the most part, have remained
unexplored. For general values of the parameters and twisted boundary
conditions the model possesses ${\rm U}(1)$ invariance. In this paper we
discuss the restrictions imposed on the parameters for which additional global
symmetries arise that are consistent with the integrable structure. These
include the lattice counterparts of ${\cal C}$, ${\cal P}$ and ${\cal T}$ as
well as translational invariance. The special properties of the lattice system
that possesses an additional ${\cal Z}_r$ invariance are considered. We also
describe the Hermitian structures, which are consistent with the integrable
one. The analysis lays the groundwork for studying the scaling limit of the
inhomogeneous six-vertex model. | math-ph |
Conformal Mappings and Dispersionless Toda hierarchy: Let $\mathfrak{D}$ be the space consists of pairs $(f,g)$, where $f$ is a
univalent function on the unit disc with $f(0)=0$, $g$ is a univalent function
on the exterior of the unit disc with $g(\infty)=\infty$ and
$f'(0)g'(\infty)=1$. In this article, we define the time variables $t_n, n\in
\Z$, on $\mathfrak{D}$ which are holomorphic with respect to the natural
complex structure on $\mathfrak{D}$ and can serve as local complex coordinates
for $\mathfrak{D}$. We show that the evolutions of the pair $(f,g)$ with
respect to these time coordinates are governed by the dispersionless Toda
hierarchy flows. An explicit tau function is constructed for the dispersionless
Toda hierarchy. By restricting $\mathfrak{D}$ to the subspace $\Sigma$ consists
of pairs where $f(w)=1/\bar{g(1/\bar{w})}$, we obtain the integrable hierarchy
of conformal mappings considered by Wiegmann and Zabrodin \cite{WZ}. Since
every $C^1$ homeomorphism $\gamma$ of the unit circle corresponds uniquely to
an element $(f,g)$ of $\mathfrak{D}$ under the conformal welding
$\gamma=g^{-1}\circ f$, the space $\text{Homeo}_{C}(S^1)$ can be naturally
identified as a subspace of $\mathfrak{D}$ characterized by $f(S^1)=g(S^1)$. We
show that we can naturally define complexified vector fields $\pa_n, n\in \Z$
on $\text{Homeo}_{C}(S^1)$ so that the evolutions of $(f,g)$ on
$\text{Homeo}_{C}(S^1)$ with respect to $\pa_n$ satisfy the dispersionless Toda
hierarchy. Finally, we show that there is a similar integrable structure for
the Riemann mappings $(f^{-1}, g^{-1})$. Moreover, in the latter case, the time
variables are Fourier coefficients of $\gamma$ and $1/\gamma^{-1}$. | math-ph |
On convergence to equilibrium for one-dimensional chain of harmonic
oscillators in the half-line: The initial-boundary value problem for an infinite one-dimensional chain of
harmonic oscillators on the half-line is considered. The large time asymptotic
behavior of solutions is studied. The initial data of the system are supposed
to be a random function which has some mixing properties. We study the
distribution $\mu_t$ of the random solution at time moments $t\in\mathbb{R}$.
The main result is the convergence of $\mu_t$ to a Gaussian probability measure
as $t\to\infty$. We find stationary states in which there is a non-zero energy
current at origin. | math-ph |
The role of curvature and stretching in dynamo plasmas: Vishik's antidynamo theorem is applied to non-stretched twisted magnetic flux
tube in Riemannian space. Marginal or slow dynamos along curved (folded),
torsioned (twisted) and non-stretching flux tubes plasma flows are obtained}.
Riemannian curvature of twisted magnetic flux tube is computed in terms of the
Frenet curvature in the thin tube limit. It is shown that, for non-stretched
filaments fast dynamo action in diffusive case cannot be obtained, in agreement
with Vishik's argument, that fast dynamo cannot be obtained in non-stretched
flows. \textbf{In this case a non-uniform stretching slow dynamo is
obtained}.\textbf{An example is given which generalizes plasma dynamo laminar
flows, recently presented by Wang et al [Phys Plasmas (2002)], in the case of
low magnetic Reynolds number $Re_{m}\ge{210}$. Curved and twisting Riemannian
heliotrons, where non-dynamo modes are found even when stretching is presented,
shows that the simple presence of stretching is not enough for the existence of
dynamo action. Folding is equivalent to Riemann curvature and can be used to
cancell magnetic fields, not enhancing the dynamo action. In this case
non-dynamo modes are found for certain values of torsion or Frenet curvature
(folding) in the spirit of anti-dynamo theorem. It is shown that curvature and
stretching are fundamental for the existence of fast dynamos in plasmas. | math-ph |
Solutions to some Molecular Potentials in D-Dimensions: Asymptotic
Iteration Method: We give a study of some molecular vibration potentials by solving the
D-dimensional Schrodinger equation using the asymptotic iteration method (AIM).
The eigenvalue values obtained by the AIM are found to agree with analytic
solutions. The corresponding eigenfunctions are also obtained using the AIM | math-ph |
Subsonic phase transition waves in bistable lattice models with small
spinodal region: Phase transitions waves in atomic chains with double-well potential play a
fundamental role in materials science, but very little is known about their
mathematical properties. In particular, the only available results about waves
with large amplitudes concern chains with piecewise-quadratic pair potential.
In this paper we consider perturbations of a bi-quadratic potential and prove
that the corresponding three-parameter family of waves persists as long as the
perturbation is small and localised with respect to the strain variable. As a
standard Lyapunov-Schmidt reduction cannot be used due to the presence of an
essential spectrum, we characterise the perturbation of the wave as a fixed
point of a nonlinear and nonlocal operator and show that this operator is
contractive in a small ball in a suitable function space. Moreover, we derive a
uniqueness result for phase transition waves with certain properties and
discuss the kinetic relation. | math-ph |
Reflection probabilities of one-dimensional Schroedinger operators and
scattering theory: The dynamic reflection probability and the spectral reflection probability
for a one-dimensional Schroedinger operator $H = - \Delta + V$ are
characterized in terms of the scattering theory of the pair $(H, H_\infty)$
where $H_\infty$ is the operator obtained by decoupling the left and right
half-lines $\mathbb{R}_{\leq 0}$ and $\mathbb{R}_{\geq 0}$. An immediate
consequence is that these reflection probabilities are in fact the same, thus
providing a short and transparent proof of the main result of Breuer, J., E.
Ryckman, and B. Simon (2010) . This approach is inspired by recent developments
in non-equilibrium statistical mechanics of the electronic black box model and
follows a strategy parallel to the Jacobi case. | math-ph |
A Loop Reversibility and Subdiffusion of the Rotor-Router Walk: The rotor-router model on a graph describes a discrete-time walk accompanied
by the deterministic evolution of configurations of rotors randomly placed on
vertices of the graph. We prove the following property: if at some moment of
time, the rotors form a closed clockwise contour on the planar graph, then the
clockwise rotations of rotors generate a walk which enters into the contour at
some vertex $v$, performs a number of steps inside the contour so that the
contour formed by rotors becomes anti-clockwise, and then leaves the contour at
the same vertex $v$. This property generalizes the previously proved theorem
for the case when the rotor configuration inside the contour is a cycle-rooted
spanning tree, and all rotors inside the contour perform a full rotation. We
use the proven property for an analysis of the sub-diffusive behavior of the
rotor-router walk. | math-ph |
On the Solution of the Van der Pol Equation: We linearize and solve the Van der Pol equation (with additional nonlinear
terms) by the application of a generalized form of Cole-Hopf transformation. We
classify also Lienard equations with low order polynomial coefficients which
can be linearized by this transformation. | math-ph |
Emergence of a singularity for Toeplitz determinants and Painleve V: We obtain asymptotic expansions for Toeplitz determinants corresponding to a
family of symbols depending on a parameter $t$. For $t$ positive, the symbols
are regular so that the determinants obey Szeg\H{o}'s strong limit theorem. If
$t=0$, the symbol possesses a Fisher-Hartwig singularity. Letting $t\to 0$ we
analyze the emergence of a Fisher-Hartwig singularity and a transition between
the two different types of asymptotic behavior for Toeplitz determinants. This
transition is described by a special Painlev\'e V transcendent. A particular
case of our result complements the classical description of Wu, McCoy, Tracy,
and Barouch of the behavior of a 2-spin correlation function for a large
distance between spins in the two-dimensional Ising model as the phase
transition occurs. | math-ph |
Low-temperature spectrum of the quantum transfer matrix of the XXZ chain
in the massless regime: The free energy per lattice site of a quantum spin chain in the thermodynamic
limit is determined by a single `dominant' Eigenvalue of an associated quantum
transfer matrix in the infinite Trotter number limit. For integrable quantum
spin chains, related with solutions of the Yang-Baxter equation, an appropriate
choice of the quantum transfer matrix enables to study its spectrum, e.g.\ by
means of the algebraic Bethe Ansatz. In its turn, the knowledge of the full
spectrum allows one to study its universality properties such as the appearance
of a conformal spectrum in the low-temperature regime. More generally,
accessing the full spectrum is a necessary step for deriving thermal form
factor series representations of the correlation functions of local operators
for the spin chain under consideration. These are statements that have been
established by physicists on a heuristic level and that are calling for a
rigorous mathematical justification. In this work we implement certain aspects
of this programme with the example of the XXZ quantum spin chain in the
antiferromagnetic massless regime and in the low-temperature limit. We
rigorously establish the existence, uniqueness and characterise the form of the
solutions to the non-linear integral equations that are equivalent to the Bethe
Ansatz equations for the quantum transfer matrix of this model. This allows us
to describe that part of the quantum transfer matrix spectrum that is related
to the Bethe Ansatz and that does not collapse to zero in the infinite Trotter
number limit. Within the considered part of the spectrum we rigorously identify
the dominant Eigenvalue and show that those correlations lengths that diverge
in the low-temperature limit are given, to the leading order, by the spectrum
of the free Boson $c=1$ conformal field theory. This rigorously establishes a
long-standing conjecture present in the physics literature. | math-ph |
Interacting fermions on the half-line: boundary counterterms and
boundary corrections: Recent years witnessed an extensive development of the theory of the critical
point in two-dimensional statistical systems, which allowed to prove {\it
existence} and {\it conformal invariance} of the {\it scaling limit} for
two-dimensional Ising model and dimers in planar graphs. Unfortunately, we are
still far from a full understanding of the subject: so far, exact solutions at
the lattice level, in particular determinant structure and exact discrete
holomorphicity, play a cucial role in the rigorous control of the scaling
limit. The few results about not-integrable (interacting) systems at
criticality are still unable to deal with {\it finite domains} and {\it
boundary corrections}, which are of course crucial for getting informations
about conformal covariance. In this thesis, we address the question of adapting
constructive Renormalization Group methods to non-integrable critical systems
in $d= 1+1$ dimensions. We study a system of interacting spinless fermions on a
one-dimensional semi-infinite lattice, which can be considered as a prototype
of the Luttinger universality class with Dirichlet Boundary Conditions. We
develop a convergent renormalized expression for the thermodynamic observables
in the presence of a quadratic {\it boundary defect} counterterm, polynomially
localized at the boundary. In particular, we get explicit bounds on the
boundary corrections to the specific ground state energy. | math-ph |
Kinetic Limit for Wave Propagation in a Random Medium: We study crystal dynamics in the harmonic approximation. The atomic masses
are weakly disordered, in the sense that their deviation from uniformity is of
order epsilon^(1/2). The dispersion relation is assumed to be a Morse function
and to suppress crossed recollisions. We then prove that in the limit epsilon
to 0 the disorder averaged Wigner function on the kinetic scale, time and space
of order epsilon^(-1), is governed by a linear Boltzmann equation. | math-ph |
Functional differentiability in time-dependent quantum mechanics: In this work we investigate the functional differentiability of the
time-dependent many-body wave function and of derived quantities with respect
to time-dependent potentials. For properly chosen Banach spaces of potentials
and wave functions Fr\'echet differentiability is proven. From this follows an
estimate for the difference of two solutions to the time-dependent
Schr\"odinger equation that evolve under the influence of different potentials.
Such results can be applied directly to the one-particle density and to bounded
operators, and present a rigorous formulation of non-equilibrium
linear-response theory where the usual Lehmann representation of the
linear-response kernel is not valid. Further, the Fr\'echet differentiability
of the wave function provides a new route towards proving basic properties of
time-dependent density-functional theory. | math-ph |
Zeta function regularization in Casimir effect calculations and J.S.
Dowker's contribution: A summary of relevant contributions, ordered in time, to the subject of
operator zeta functions and their application to physical issues is provided.
The description ends with the seminal contributions of Stephen Hawking and
Stuart Dowker and collaborators, considered by many authors as the actual
starting point of the introduction of zeta function regularization methods in
theoretical physics, in particular, for quantum vacuum fluctuation and Casimir
effect calculations. After recalling a number of the strengths of this powerful
and elegant method, some of its limitations are discussed. Finally, recent
results of the so called operator regularization procedure are presented. | math-ph |
The beta-Hermite and beta-Laguerre processes: In this work, we introduce matrix-valued diffusion processes which describe
the non-equilibrium situation of the matrix models for the beta-Hermite and the
beta-Laguerre ensembles. We also study the corresponding spectral measure
process and empirical/singular value process with regard to their limit laws. | math-ph |
Quantum Trajectories, State Diffusion and Time Asymmetric Eventum
Mechanics: We show that the quantum stochastic unitary dynamics Langevin model for
continuous in time measurements provides an exact formulation of the Heisenberg
uncertainty error-disturbance principle. Moreover, as it was shown in the 80's,
this Markov model induces all stochastic linear and non-linear equations of the
phenomenological "quantum trajectories" such as quantum state diffusion and
spontaneous localization by a simple quantum filtering method. Here we prove
that the quantum Langevin equation is equivalent to a Dirac type boundary-value
problem for the second-quantized input "offer waves from future" in one extra
dimension, and to a reduction of the algebra of the consistent histories of
past events to an Abelian subalgebra for the "trajectories of the output
particles". This result supports the wave-particle duality in the form of the
thesis of Eventum Mechanics that everything in the future is constituted by
quantized waves, everything in the past by trajectories of the recorded
particles. We demonstrate how this time arrow can be derived from the principle
of quantum causality for nondemolition continuous in time measurements. | math-ph |
II - Conservation of Gravitational Energy Momentum and
Poincare-Covariant Classical Theory of Gravitation: Viewing gravitational energy-momentum $p_G^\mu$ as equal by observation, but
different in essence from inertial energy-momentum $p_I^\mu$ naturally leads to
the gauge theory of volume-preserving diffeormorphisms of an inner Minkowski
space ${\bf M}^{\sl 4}$. To extract its physical content the full gauge group
is reduced to its Poincar\'e subgroup. The respective Poincar\'e gauge fields,
field strengths and Poincar\'e-covariant field equations are obtained and
point-particle source currents are derived. The resulting set of non-linear
field equations coupled to point matter is solved in first order resulting in
Lienard-Wiechert-like potentials for the Poincar\'e fields. After numerical
identification of gravitational and inertial energy-momentum Newton's inverse
square law for gravity in the static non-relativistic limit is recovered. The
Weak Equivalence Principle in this approximation is proven to be valid and
spacetime geometry in the presence of Poincar\'e fields is shown to be curved.
Finally, the gravitational radiation of an accelerated point particle is
calulated. | math-ph |
The Dry Ten Martini Problem at Criticality: We apply recently developed methods for the construction of quasi-periodic
transfer matrices to the Dry Ten Martini problem for the critical
almost-Mathieu Operator, also known as the Aubry-Andre-Harper (AAH) model. | math-ph |
Invariant Properties of the Ansatz of the Hirota Method for Quasilinear
Parabolic equations: We propose a new method based on the invariant properties of the ansatz of
the Hirota method which have been discovered recently. This method allows one
to construct new solutions for a certain class the dissipative equations
classified by degrees of homogeneity. This algorithm is similar to the method
of ``dressing'' the solutions of integrable equations. A class of new solutions
is constructed. It is proved that all known exact solutions of the
FitzHygh-Nagumo-Semenov equation can be expressed in terms of solutions of the
linear parabolic equation. This method is compared with the Miura transforms in
the theory of Kortveg de Vris equations. This method allows on to create a
package by using the methods of computer algebra. | math-ph |
The JLO Character for The Noncommutative Space of Connections of
Aastrup-Grimstrup-Nest: In attempts to combine non-commutative geometry and quantum gravity,
Aastrup-Grimstrup-Nest construct a semi-finite spectral triple, modeling the
space of G-connections for G=U(1) or SU(2). AGN show that the interaction
between the algebra of holonomy loops and the Dirac-type operator D reproduces
the Poisson structure of General Relativity in Ashtekar's loop variables. This
article generalizes AGN's construction to any connected compact Lie group G. A
construction of AGN's semi-finite spectral triple in terms of an inductive
limit of spectral triples is formulated. The refined construction permits the
semi-finite spectral triple to be even when G is even dimensional. The
Dirac-type operator D in AGN's semi-finite spectral triple is a weighted sum of
a basic Dirac operator on G. The weight assignment is a diverging sequence that
governs the "volume" associated to each copy of G. The JLO cocycle of AGN's
triple is examined in terms of the weight assignment. An explicit condition on
the weight assignment perturbations is given, so that the associated JLO class
remains invariant. Such a condition leads to a functoriality property of AGN's
construction. | math-ph |
Crossing Probabilities and Modular Forms: We examine crossing probabilities and free energies for conformally invariant
critical 2-D systems in rectangular geometries, derived via conformal field
theory and Stochastic L\"owner Evolution methods. These quantities are shown to
exhibit interesting modular behavior, although the physical meaning of modular
transformations in this context is not clear. We show that in many cases these
functions are completely characterized by very simple transformation
properties. In particular, Cardy's function for the percolation crossing
probability (including the conformal dimension 1/3), follows from a simple
modular argument. A new type of "higher-order modular form" arises and its
properties are discussed briefly. | math-ph |
Universality of the momentum band density of periodic networks: The momentum spectrum of a periodic network (quantum graph) has a band-gap
structure. We investigate the relative density of the bands or, equivalently,
the probability that a randomly chosen momentum belongs to the spectrum of the
periodic network. We show that this probability exhibits universal properties.
More precisely, the probability to be in the spectrum does not depend on the
edge lengths (as long as they are generic) and is also invariant within some
classes of graph topologies. | math-ph |
Expectation variables on a para-contact metric manifold exactly derived
from master equations: Based on information and para-contact metric geometries, in this paper a
class of dynamical systems is formulated for describing time-development of
expectation variables. Here such systems for expectation variables are exactly
derived from continuous-time master equations describing nonequilibrium
processes. | math-ph |
New series of multi-parametric solutions to GYBE: quantum gates and
integrability: We obtain two series of spectral parameter dependent solutions to the
generalized Yang-Baxter equations (GYBE), for definite types of $N_1^2\times
N_2^2$ matrices with general dimensions $N_1$ and $N_2$. Appropriate extensions
are presented for the inhomogeneous GYBEs. The first series of the solutions
includes as particular cases the $X$-shaped trigonometric braiding matrices.
For construction of the second series the colored and graded permutation
operators are defined, and multi-spectral parameter Yang-Baxterization is
performed. For some examples the corresponding integrable models are discussed.
The unitary solutions existing in these two series can be considered as
generalizations of the multipartite Bell matrices in the quantum information
theory. | math-ph |
Decay of Superconducting Correlations for Gauged Electrons in Dimensions
$D\le 4$: We study lattice superconductors coupled to gauge fields, such as an
attractive Hubbard model in electromagnetic fields, with a standard gauge
fixing. We prove upper bounds for a two-point Cooper pair correlation at finite
temperatures in spatial dimensions $D\le 4$. The upper bounds decay
exponentially in three dimensions, and by power law in four dimensions. These
imply absence of the superconducting long-range order for the Cooper pair
amplitude as a consequence of fluctuations of the gauge fields. Since our
results hold for the gauge fixing Hamiltonian, they cannot be obtained as a
corollary of Elitzur's theorem. | math-ph |
Integrable hierarchies associated to infinite families of Frobenius
manifolds: We propose a new construction of an integrable hierarchy associated to any
infinite series of Frobenius manifolds satisfying a certain stabilization
condition. We study these hierarchies for Frobenius manifolds associated to
$A_N$, $D_N$ and $B_N$ singularities. In the case of $A_N$ Frobenius manifolds
our hierarchy turns out to coincide with the KP hierarchy; for $B_N$ Frobenius
manifolds it coincides with the BKP hierarchy; and for $D_N$ hierarchy it is a
certain reduction of the 2-component BKP hierarchy. As a side product to these
results we illustrate the enumerative meaning of certain coefficients of $A_N$,
$D_N$ and $B_N$ Frobenius potentials. | math-ph |
Eigenvalue asymptotics for the damped wave equation on metric graphs: We consider the linear damped wave equation on finite metric graphs and
analyse its spectral properties with an emphasis on the asymptotic behaviour of
eigenvalues. In the case of equilateral graphs and standard coupling conditions
we show that there is only a finite number of high-frequency abscissas, whose
location is solely determined by the averages of the damping terms on each
edge. We further describe some of the possible behaviour when the edge lengths
are no longer necessarily equal but remain commensurate. | math-ph |
Matrix Field Theory: This thesis studies matrix field theories, which are a special type of matrix
models. First, the different types of applications are pointed out, from
(noncommutative) quantum field theory over 2-dimensional quantum gravity up to
algebraic geometry with explicit computation of intersection numbers on the
moduli space of complex curves.
The Kontsevich model, which has proved the Witten conjecture, is the simplest
example of a matrix field theory. Generalisations of this model will be
studied, where different potentials and the spectral dimension (defined by the
asymptotics of the external matrix) are introduced. Because they are naturally
embedded into a Riemann surface, the correlation functions are graded by the
genus and the number of boundary components. The renormalisation procedure of
quantum field theory leads to finite UV-limit.
We provide a method to determine closed Schwinger-Dyson equations with the
usage of Ward-Takahashi identities in the continuum limit. The cubic
(Kontsevich model) and the quartic (Grosse-Wulkenhaar model) potentials are
studied separately. For the cubic potential, we show that the renormalisation
procedure is compatible with topological recursion (TR). This means that the
exact results computed by TR coincide perturbatively with the graph expansion
renormalised by Zimmermann's forest formula. For the quartic model, the first
correlation function (2-point function) is computed exactly. We give hints that
the quartic model has structurally the same properties as the hermitian
2-matrix model with genus zero spectral curve. | math-ph |
Characterization and solvability of quasipolynomial symplectic mappings: Quasipolynomial (or QP) mappings constitute a wide generalization of the
well-known Lotka-Volterra mappings, of importance in different fields such as
population dynamics, Physics, Chemistry or Economy. In addition, QP mappings
are a natural discrete-time analog of the continuous QP systems, which have
been extensively used in different pure and applied domains. After presenting
the basic definitions and properties of QP mappings in a previous article
\cite{bl1}, the purpose of this work is to focus on their characterization by
considering the existence of symplectic QP mappings. In what follows such QP
symplectic maps are completely characterized. Moreover, use of the QP formalism
can be made in order to demonstrate that all QP symplectic mappings have an
analytical solution that is explicitly and generally constructed. Examples are
given. | math-ph |
Heisenberg Groups in the Theory of the Lattice Peierls Electron: the
Irrational Flux Case: It is shown that the quantum mechanics of a charged particle moving in a
uniform magnetic field in the plane (Landau) or on a planar lattice (Peierls)
is described in all detail by the projective representation theory of the
"euclidean" group of the appropriate configuration space. In the Landau case, a
detailed description of the state space as well as the determination of the
correct Hamiltonian follows from the properties of the real Heisenberg group,
especially the fact that it has an essentially unique irreducible
representation. In the Peierls case, the corresponding groups are infinite
discrete translation groups centrally extended by the circle group. For
irrational flux/plaquette (in units of the flux quantum) these groups are
"almost Heisenberg" in the sense that they have a distinguished irreducible
representation which plays, in the quantum theory, the role of the unique
representation of the real Heisenberg group. The physics is fully determined
by, and is periodic in, the value of the flux/plaquette. The Hamiltonian for
nearest neighbour hopping is the Harper Hamiltonian. Vector potentials are not
introduced. | math-ph |
Trace formulae for Schrodinger operators on metric graphs with
applications to recovering matching conditions: The paper is a continuation of the study started in \cite{Yorzh1}.
Schrodinger operators on finite compact metric graphs are considered under the
assumption that the matching conditions at the graph vertices are of $\delta$
type. Either an infinite series of trace formulae (provided that edge
potentials are infinitely smooth) or a finite number of such formulae (in the
cases of $L_1$ and $C^M$ edge potentials) are obtained which link together two
different quantum graphs under the assumption that their spectra coincide.
Applications are given to the problem of recovering matching conditions for a
quantum graph based on its spectrum. | math-ph |
Spectrum of Lebesgue measure zero for Jacobi matrices of quasicrystals: We study one-dimensional random Jacobi operators corresponding to strictly
ergodic dynamical systems. In this context, we characterize the spectrum of
these operators by non-uniformity of the transfer matrices and the set where
the Lyapunov exponent vanishes. Adapting this result to subshifts satisfying
the so-called Boshernitzan condition, it turns out that the spectrum is
supported on a Cantor set with Lebesgue measure zero. This generalizes earlier
results for Schr\"odinger operators. | math-ph |
New differential equations in the six-vertex model: This letter is concerned with the analysis of the six-vertex model with
domain-wall boundaries in terms of partial differential equations (PDEs). The
model's partition function is shown to obey a system of PDEs resembling the
celebrated Knizhnik-Zamolodchikov equation. The analysis of our PDEs naturally
produces a family of novel determinant representations for the model's
partition function. | math-ph |
A geometric study of many body systems: A n n-body system is a labelled collection of n point masses in Euclidean
space, and their congruence and internal symmetry properties involve a rich
mathematical structure which is investigated in the framework of equivariant
Riemannian geometry. Some basic concepts are n-configuration, configuration
space, internal space, shape space, Jacobi transformations and weighted root
system. The latter is a generalization of the root system of SU(n), which
provides a bookkeeping for expressing the mutual distances of the point masses
in terms of the Jacobi vectors. Moreover, its application to the study of
collinear central n-configurations yields a simple proof of Moulton's
enumeration formula. A major topic is the general study of matrix spaces
representing the shape space of many body systems in Euclidean k-space, the
structure of the m-universal shape space and its O(m)-equivariant linear
model.This also leads to those orbital fibrations where SO(m) or O(m) act on a
sphere with a sphere as orbit space. Some examples of this kind are encountered
in the literature, e.g. the special case of the 5-sphere mod O(2), which equals
the 4-sphere, was analyzed independently by Arnold, Kuiper and Massey in the
1970's. | math-ph |
Multisolitonic solutions from a Bäcklund transformation for a
parametric coupled Korteweg-de Vries system: We introduce a parametric coupled KdV system which contains, for particular
values of the parameter, the complex extension of the KdV equation and one of
the Hirota-Satsuma integrable systems. We obtain a generalized Gardner
transformation and from the associated $\varepsilon$- deformed system we get
the infinite sequence of conserved quantities for the parametric coupled
system. We also obtain a B\"{a}cklund transformation for the system. We prove
the associated permutability theorem corresponding to such transformation and
we generate new multi-solitonic and periodic solutions for the system depending
on several parameters. We show that for a wide range of the parameters the
solutions obtained from the permutability theorem are regular solutions.
Finally we found new multisolitonic solutions propagating on a non-trivial
regular static background. | math-ph |
Multicritical continuous random trees: We introduce generalizations of Aldous' Brownian Continuous Random Tree as
scaling limits for multicritical models of discrete trees. These discrete
models involve trees with fine-tuned vertex-dependent weights ensuring a k-th
root singularity in their generating function. The scaling limit involves
continuous trees with branching points of order up to k+1. We derive explicit
integral representations for the average profile of this k-th order
multicritical continuous random tree, as well as for its history distributions
measuring multi-point correlations. The latter distributions involve
non-positive universal weights at the branching points together with fractional
derivative couplings. We prove universality by rederiving the same results
within a purely continuous axiomatic approach based on the resolution of a set
of consistency relations for the multi-point correlations. The average profile
is shown to obey a fractional differential equation whose solution involves
hypergeometric functions and matches the integral formula of the discrete
approach. | math-ph |
Congruence method for global Darboux reduction of finite-dimensional
Poisson systems: A new procedure for the global construction of the Casimir invariants and
Darboux canonical form for finite-dimensional Poisson systems is developed.
This approach is based on the concept of matrix congruence and can be applied
without the previous determination of the Casimir invariants (recall that their
prior knowledge is unavoidable for the standard reduction methods, thus
requiring either the integration of a system of PDEs or solving some equivalent
problem). Well the opposite, in the new congruence method, both the Darboux
coordinates and the Casimir invariants arise simultaneously as the outcome of
the reduction algorithm. In fact, the congruence algorithm proceeds only in
terms of matrix-algebraic transformations and direct quadratures, thus avoiding
the need of previously integrating a system of PDEs and therefore improving
previously known approaches. Physical examples illustrating different aspects
of the theory are provided. | math-ph |
Derivation of an eigenvalue probability density function relating to the
Poincare disk: A result of Zyczkowski and Sommers [J.Phys.A, 33, 2045--2057 (2000)] gives
the eigenvalue probability density function for the top N x N sub-block of a
Haar distributed matrix from U(N+n). In the case n \ge N, we rederive this
result, starting from knowledge of the distribution of the sub-blocks,
introducing the Schur decomposition, and integrating over all variables except
the eigenvalues. The integration is done by identifying a recursive structure
which reduces the dimension. This approach is inspired by an analogous approach
which has been recently applied to determine the eigenvalue probability density
function for random matrices A^{-1} B, where A and B are random matrices with
entries standard complex normals. We relate the eigenvalue distribution of the
sub-blocks to a many body quantum state, and to the one-component plasma, on
the pseudosphere. | math-ph |
Harmonic factorization and reconstruction of the elasticity tensor: In this paper, we propose a factorization of a fourth-order harmonic tensor
into second-order tensors. We obtain moreover explicit equivariant
reconstruction formulas, using second-order covariants, for transverse
isotropic and orthotropic harmonic fourth-order tensors, and for trigonal and
tetragonal harmonic fourth-order tensors up to a cubic fourth order covariant
remainder. | math-ph |
Baker-Akhiezer functions and generalised Macdonald-Mehta integrals: For the rational Baker-Akhiezer functions associated with special
arrangements of hyperplanes with multiplicities we establish an integral
identity, which may be viewed as a generalisation of the self-duality property
of the usual Gaussian function with respect to the Fourier transformation. We
show that the value of properly normalised Baker-Akhiezer function at the
origin can be given by an integral of Macdonald-Mehta type and explicitly
compute these integrals for all known Baker-Akhiezer arrangements. We use the
Dotsenko-Fateev integrals to extend this calculation to all deformed root
systems, related to the non-exceptional basic classical Lie superalgebras. | math-ph |
Abrupt Convergence and Escape Behavior for Birth and Death Chains: We link two phenomena concerning the asymptotical behavior of stochastic
processes: (i) abrupt convergence or cut-off phenomenon, and (ii) the escape
behavior usually associated to exit from metastability. The former is
characterized by convergence at asymptotically deterministic times, while the
convergence times for the latter are exponentially distributed. We compare and
study both phenomena for discrete-time birth-and-death chains on Z with drift
towards zero. In particular, this includes energy-driven evolutions with energy
functions in the form of a single well. Under suitable drift hypotheses, we
show that there is both an abrupt convergence towards zero and escape behavior
in the other direction. Furthermore, as the evolutions are reversible, the law
of the final escape trajectory coincides with the time reverse of the law of
cut-off paths. Thus, for evolutions defined by one-dimensional energy wells
with sufficiently steep walls, cut-off and escape behavior are related by time
inversion. | math-ph |
Real gas flows issued from a source: Stationary adiabatic flows of real gases issued from a source of given
intensity are studied. Thermodynamic states of gases are described by
Legendrian or Lagrangian manifolds. Solutions of Euler equations are given
implicitly for any equation of state and the behavior of solutions of the
Navier-Stokes equations with the viscosity considered as a small parameter is
discussed. For different intensities of the source we introduce a small
parameter into the Navier-Stokes equation and construct corresponding
asymptotic expansions. We consider the most popular model of real gases --- the
van der Waals model, and ideal gases as well. | math-ph |
Superintegrable Extensions of Superintegrable Systems: A procedure to extend a superintegrable system into a new superintegrable one
is systematically tested for the known systems on $\mathbb E^2$ and $\mathbb
S^2$ and for a family of systems defined on constant curvature manifolds. The
procedure results effective in many cases including
Tremblay-Turbiner-Winternitz and three-particle Calogero systems. | math-ph |
Copositivity for 3rd order symmetric tensors and applications: The strict opositivity of 4th order symmetric tensor may apply to detect
vacuum stability of general scalar potential. For finding analytical
expressions of (strict) opositivity of 4th order symmetric tensor, we may
reduce its order to 3rd order to better deal with it. So, it is provided that
several analytically sufficient conditions for the copositivity of 3th order 2
dimensional (3 dimensional) symmetric tensors. Subsequently, applying these
conclusions to 4th order tensors, the analytically sufficient conditions of
copositivity are proved for 4th order 2 dimensional and 3 dimensional symmetric
tensors. Finally, we apply these results to present analytical vacuum stability
conditions for vacuum stability for $\mathbb{Z}_3$ scalar dark matter. | math-ph |
On the relaxation rate of short chains of rotors interacting with
Langevin thermostats: In this short note, we consider a system of two rotors, one of which
interacts with a Langevin heat bath. We show that the system relaxes to its
invariant measure (steady state) no faster than a stretched exponential
$\exp(-c t^{1/2})$. This indicates that the exponent $1/2$ obtained earlier by
the present authors and J.-P. Eckmann for short chains of rotors is optimal. | math-ph |
Multiple scaling limits of $\mathrm{U}(N)^2 \times \mathrm{O}(D)$
multi-matrix models: We study the double- and triple-scaling limits of a complex multi-matrix
model, with $\mathrm{U}(N)^2\times \mathrm{O}(D)$ symmetry. The double-scaling
limit amounts to taking simultaneously the large-$N$ (matrix size) and
large-$D$ (number of matrices) limits while keeping the ratio $N/\sqrt{D}=M$
fixed. The triple-scaling limit consists in taking the large-$M$ limit while
tuning the coupling constant $\lambda$ to its critical value $\lambda_c$ and
keeping fixed the product $M(\lambda_c-\lambda)^\alpha$, for some value of
$\alpha$ that depends on the particular combinatorial restrictions imposed on
the model. Our first main result is the complete recursive characterization of
the Feynman graphs of arbitrary genus which survive in the double-scaling
limit. Next, we classify all the dominant graphs in the triple-scaling limit,
which we find to have a plane binary tree structure with decorations. Their
critical behavior belongs to the universality class of branched polymers.
Lastly, we classify all the dominant graphs in the triple-scaling limit under
the restriction to three-edge connected (or two-particle irreducible) graphs.
Their critical behavior falls in the universality class of Liouville quantum
gravity (or, in other words, the Brownian sphere). | math-ph |
CleGo: A package for automated computation of Clebsch-Gordan
coefficients in Tensor Product Representations for Lie Algebras A - G: We present a program that allows for the computation of tensor products of
irreducible representations of Lie algebras A-G based on the explicit
construction of weight states. This straightforward approach (which is slower
and more memory-consumptive than the standard methods to just calculate
dimensions of the tensor product decomposition) produces Clebsch-Gordan
coefficients that are of interest for instance in discussing symmetry breaking
in model building for grand unified theories. For that purpose, multiple tensor
products have been implemented as well as means for analyzing the resulting
effective operators in particle physics. | math-ph |
Weakly bound states in heterogeneous waveguides: We study the spectrum of the Helmholtz equation in a two-dimensional infinite
waveguide, containing a weak heterogeneity localized at an internal point, and
obeying Dirichlet boundary conditions at its border. We prove that, when the
heterogeneity corresponds to a locally denser material, the lowest eigenvalue
of the spectrum falls below the continuum threshold and a bound state appears,
localized at the heterogeneity. We devise a rigorous perturbation scheme and
derive the exact expression for the energy to third order in the heterogeneity. | math-ph |
On the emergence of quantum Boltzmann fluctuation dynamics near a
Bose-Einstein Condensate: In this work, we study the quantum fluctuation dynamics in a Bose gas on a
torus $\Lambda=(L\mathbb{T})^3$ that exhibits Bose-Einstein condensation,
beyond the leading order Hartree-Fock-Bogoliubov (HFB) fluctuations. Given a
Bose-Einstein condensate (BEC) with density $N$ surrounded by thermal
fluctuations with density $1$, we assume that the system is described by a
mean-field Hamiltonian. We extract a quantum Boltzmann type dynamics from a
second-order Duhamel expansion upon subtracting both the BEC dynamics and the
HFB dynamics. Using a Fock-space approach, we provide explicit error bounds. It
is known that the BEC and the HFB fluctuations both evolve at microscopic time
scales $t\sim1$. Given a quasifree initial state, we determine the time
evolution of the centered correlation functions $\langle a\rangle$, $\langle
aa\rangle-\langle a\rangle^2$, $\langle a^+a\rangle-|\langle a\rangle|^2$ at
mesoscopic time scales $t\sim\lambda^{-2}$, where $0<\lambda\ll1$ denotes the
size of the HFB interaction. For large but finite $N$, we consider both the
case of fixed system size $L\sim1$, and the case $L\sim \lambda^{-2-}$. In the
case $L\sim1$, we show that the Boltzmann collision operator contains
subleading terms that can become dominant, depending on time-dependent
coefficients assuming particular values in $\mathbb{Q}$; this phenomenon is
reminiscent of the Talbot effect. For the case $L\sim \lambda^{-2-}$, we prove
that the collision operator is well approximated by the expression predicted in
the literature. In either of those cases, we have $\lambda\sim \Big(\frac{\log
\log N}{\log N}\Big)^{\alpha}$, for different values of $\alpha>0$. | math-ph |
Critical Topology for Optimization on the Symplectic Group: Optimization problems over compact Lie groups have been extensively studied
due to their broad applications in linear programming and optimal control. This
paper analyzes least square problems over a noncompact Lie group, the
symplectic group $\Sp(2N,\R)$, which can be used to assess the optimality of
control over dynamical transformations in classical mechanics and quantum
optics. The critical topology for minimizing the Frobenius distance from a
target symplectic transformation is solved. It is shown that the critical
points include a unique local minimum and a number of saddle points. The
topology is more complicated than those of previously studied problems on
compact Lie groups such as the orthogonal and unitary groups because the
incompatibility of the Frobenius norm with the pseudo-Riemannian structure on
the symplectic group brings significant nonlinearity to the problem.
Nonetheless, the lack of traps guarantees the global convergence of local
optimization algorithms. | math-ph |
Hidden Q-structure and Lie 3-algebra for non-abelian superconformal
models in six dimensions: We disclose the mathematical structure underlying the gauge field sector of
the recently constructed non-abelian superconformal models in six spacetime
dimensions. This is a coupled system of 1-form, 2-form, and 3-form gauge
fields. We show that the algebraic consistency constraints governing this
system permit to define a Lie 3-algebra, generalizing the structural Lie
algebra of a standard Yang-Mills theory to the setting of a higher bundle.
Reformulating the Lie 3-algebra in terms of a nilpotent degree 1 BRST-type
operator Q, this higher bundle can be compactly described by means of a
Q-bundle; its fiber is the shifted tangent of the Q-manifold corresponding to
the Lie 3-algebra and its base the odd tangent bundle of spacetime equipped
with the de Rham differential. The generalized Bianchi identities can then be
retrieved concisely from Q^2=0, which encode all the essence of the structural
identities. Gauge transformations are identified as vertical inner
automorphisms of such a bundle, their algebra being determined from a Q-derived
bracket. | math-ph |
On the relation between the Maxwell system and the Dirac equation: A simple relation between the Maxwell system and the Dirac equation based on
their quaternionic reformulation is discussed. We establish a close connection
between solutions of both systems as well as a relation between the wave
parameters of the electromagnetic field and the energy of the Dirac particle. | math-ph |
A general class of invariant diffusion processes in one dimension: This paper improves a previously established test involving only coefficients
to decide a priori whether or not non-trivial symmetries of a large class of
space-time dependent diffusion processes on the real line exist. When the
existence of these symmetries are ensured, the transformation to canonical
forms admitting either four- or six-dimensional symmetry groups and the full
list of their infinitesimal generators are then immediately at our disposal
without any cumbersome calculations that happens when at least one of the
coefficients is arbitrarily chosen. We study in depth symmetry and reducibility
properties and physically important solutions of six models arising in
applications. | math-ph |
A Poisson Algebra for Abelian Yang-Mills Fields on Riemannian Manifolds
with Boundary: We define a family of observables for abelian Yang-Mills fields associated to
compact regions $U \subseteq M$ with smooth boundary in Riemannian manifolds.
Each observable is parametrized by a first variation of solutions and arises as
the integration of gauge invariant conserved current along admissible
hypersurfaces contained in the region. The Poisson bracket uses the integration
of a canonical presymplectic current. | math-ph |
Matrix mechanics of the relativistic point particle and string in
Clifford space: We resolve the space-time canonical variables of the relativistic point
particle into inner products of Weyl spinors with components in a Clifford
algebra and find that these spinors themselves form a canonical system with
generalized Poisson brackets. For N particles, the inner products of their
Clifford coordinates and momenta form two NxN Hermitian matrices X and P which
transform under a U(N) symmetry in the generating algebra. This is used as a
starting point for defining matrix mechanics for a point particle in Clifford
space. Next we consider the string. The Lorentz metric induces a metric and a
scalar on the world sheet which we represent by a Jackiw-Teitelboim term in the
action. The string is described by a polymomenta canonical system and we find
the wave solutions to the classical equations of motion for a flat world sheet.
Finally, we show that the SL(2.C) charge and space-time momentum of the
quantized string satisfy the Poincare algebra. | math-ph |
Asymptotic stability of N-solitons of the FPU lattices: We study stability of N-soliton solutions of the FPU lattice equation.
Solitary wave solutions of FPU cannot be characterized as a critical point of
conservation laws due to the lack of infinitesimal invariance in the spatial
variable. In place of standard variational arguments for Hamiltonian systems,
we use an exponential stability property of the linearized FPU equation in a
weighted space which is biased in the direction of motion.
The dispersion of the linearized FPU equation balances the potential term for
low frequencies, whereas the dispersion is superior for high frequencies. We
approximate the low frequency part of a solution of the linearized FPU equation
by a solution to the linearized KdV equation around an N-soliton.
We prove an exponential stability property of the linearized KdV equation
around N-solitons by using the linearized Backlund transformation and use the
result to analyze the linearized FPU equation. | math-ph |
Hydrodynamics of a driven lattice gas with open boundaries: the
asymmetric simple exclusion: We consider the asymmetric simple exclusion process in $d\ge 3$ with open
boundaries. The particle reservoirs of constant densities are modeled by birth
and death processes at the boundary. We prove that, if the initial density and
the densities of the boundary reservoirs differ for order of $\epsilon$ from
1/2, the density empirical field, rescaled as $\epsilon^{-1}$, converges to the
solution of the initial-boundary value problem for the viscous Burgers equation
in a finite domain with given density on the boundary. | math-ph |
Isometrodynamics and Gravity: Isometrodynamics (ID), the gauge theory of the group of volume-preserving
diffeomorphisms of an "inner" D-dimensional flat space, is tentatively
interpreted as a fundamental theory of gravity. Dimensional analysis shows that
the Planck length l_P - and through it \hbar and \Gamma - enters the gauge
field action linking ID and gravity in a natural way. Noting that the ID gauge
field couples solely through derivatives acting on "inner" space variables all
ID fields are Taylor-expanded in "inner" space. Integrating out the "inner"
space variables yields an effective field theory for the coefficient fields
with l_P^2 emerging as the expansion parameter. For \hbar goint to zero only
the leading order field does not vanish. This classical field couples to the
matter Noether currents and charges related to the translation invariance in
"inner" space. A model coupling this leading order field to a matter point
source is established and solved. Interpreting the matter Noether charge in
terms of gravitational mass Newton's inverse square law is finally derived for
a static gauge field source and a slowly moving test particle. Gravity emerges
as potentially related to field variations over "inner" space and might
microscopically be described by the ID gauge field or equivalently by an
infinite string of coefficient fields only the leading term of which is related
to the macroscopical effects of gravity. | math-ph |
Heisenberg Picture Approach to the Stability of Quantum Markov Systems: Quantum Markovian systems, modeled as unitary dilations in the quantum
stochastic calculus of Hudson and Parthasarathy, have become standard in
current quantum technological applications. This paper investigates the
stability theory of such systems. Lyapunov-type conditions in the Heisenberg
picture are derived in order to stabilize the evolution of system operators as
well as the underlying dynamics of the quantum states. In particular, using the
quantum Markov semigroup associated with this quantum stochastic differential
equation, we derive sufficient conditions for the existence and stability of a
unique and faithful invariant quantum state. Furthermore, this paper proves the
quantum invariance principle, which extends the LaSalle invariance principle to
quantum systems in the Heisenberg picture. These results are formulated in
terms of algebraic constraints suitable for engineering quantum systems that
are used in coherent feedback networks. | math-ph |
Ladder operators and coherent states for multi-step supersymmetric
rational extensions of the truncated oscillator: We construct ladder operators, $\tilde{C}$ and $\tilde{C^\dagger}$, for a
multi-step rational extension of the harmonic oscillator on the half plane,
$x\ge0$. These ladder operators connect all states of the spectrum in only
infinite-dimensional representations of their polynomial Heisenberg algebra.
For comparison, we also construct two different classes of ladder operator
acting on this system that form finite-dimensional as well as
infinite-dimensional representations of their respective polynomial Heisenberg
algebras. For the rational extension, we construct the position wavefunctions
in terms of exceptional orthogonal polynomials. For a particular choice of
parameters, we construct the coherent states, eigenvectors of $\tilde{C}$ with
generally complex eigenvalues, $z$, as superpositions of a subset of the energy
eigenvectors. Then we calculate the properties of these coherent states,
looking for classical or non-classical behaviour. We calculate the energy
expectation as a function of $|z|$. We plot position probability densities for
the coherent states and for the even and odd cat states formed from these
coherent states. We plot the Wigner function for a particular choice of $z$.
For these coherent states on one arm of a beamsplitter, we calculate the two
excitation number distribution and the linear entropy of the output state. We
plot the standard deviations in $x$ and $p$ and find no squeezing in the regime
considered. By plotting the Mandel $Q$ parameter for the coherent states as a
function of $|z|$, we find that the number statistics is sub-Poissonian. | math-ph |
The generalized solutions of the Lama's equations in the case of running
loads. The shock waves: The system of Lama's equations is investigated, describing the motion of the
elastic media under subsonic, transonic and supersonic velocities of the moving
source of distributions, and its decisions in space of generalized
vector-functions. The questions are considered connected with arising shock
waves, which appear in ambience under supersonic source of distributions. On
base of the generalized functions theories the method of the determination of
the conditions on gaps of the decisions and their derivatives on shock waves
fronts is offered. | math-ph |
Singular perturbations and Lindblad-Kossakowski differential equations: We consider an ensemble of quantum systems whose average evolution is
described by a density matrix, solution of a Lindblad-Kossakowski differential
equation. We focus on the special case where the decoherence is only due to a
highly unstable excited state and where the spontaneously emitted photons are
measured by a photo-detector. We propose a systematic method to eliminate the
fast and asymptotically stable dynamics associated to the excited state in
order to obtain another differential equation for the slow part. We show that
this slow differential equation is still of Lindblad-Kossakowski type, that the
decoherence terms and the measured output depend explicitly on the amplitudes
of quasi-resonant applied field, i.e., the control. Beside a rigorous proof of
the slow/fast (adiabatic) reduction based on singular perturbation theory, we
also provide a physical interpretation of the result in the context of
coherence population trapping via dark states and decoherence-free subspaces.
Numerical simulations illustrate the accuracy of the proposed approximation for
a 5-level systems. | math-ph |
Finding Non-Zero Stable Fixed Points of the Weighted Kuramoto model is
NP-hard: The Kuramoto model when considered over the full space of phase angles
[$0,2\pi$) can have multiple stable fixed points which form basins of
attraction in the solution space. In this paper we illustrate the fundamentally
complex relationship between the network topology and the solution space by
showing that determining the possibility of multiple stable fixed points from
the network topology is NP-hard for the weighted Kuramoto Model. In the case of
the unweighted model this problem is shown to be at least as difficult as a
number partition problem, which we conjecture to be NP-hard. We conclude that
it is unlikely that stable fixed points of the Kuramoto model can be
characterized in terms of easily computable network invariants. | math-ph |
Ground state of Bose gases interacting through singular potentials: We consider a system of $N$ bosons on the three-dimensional unit torus. The
particles interact through repulsive pair interactions of the form
$N^{3\beta-1} v (N^\beta x)$ for $\beta \in (0,1)$. We prove the next order
correction to Bogoliubov theory for the ground state and the ground state
energy. | math-ph |
Particle Trajectories for Quantum Maps: We study the trajectories of a semiclassical quantum particle under repeated
indirect measurement by Kraus operators, in the setting of the quantized torus.
In between measurements, the system evolves via either Hamiltonian propagators
or metaplectic operators. We show in both cases the convergence in total
variation of the quantum trajectory to its corresponding classical trajectory,
as defined by propagation of a semiclassical defect measure. This convergence
holds up to the Ehrenfest time of the classical system, which is larger when
the system is less chaotic. In addition, we present numerical simulations of
these effects.
In proving this result, we provide a characterization of a type of
semi-classical defect measure we call uniform defect measures. We also prove
derivative estimates of a function composed with a flow on the torus. | math-ph |
Wulff construction in statistical mechanics and in combinatorics: We present the geometric solutions to some variational problems of
statistical mechanics and combinatorics. Together with the Wulff construction,
which predicts the shape of the crystals, we discuss the construction which
exhibits the shape of a typical Young diagram and of a typical skyscraper. | math-ph |
More on Rotations as Spin Matrix Polynomials: Any nonsingular function of spin j matrices always reduces to a matrix
polynomial of order 2j. The challenge is to find a convenient form for the
coefficients of the matrix polynomial. The theory of biorthogonal systems is a
useful framework to meet this challenge. Central factorial numbers play a key
role in the theoretical development. Explicit polynomial coefficients for
rotations expressed either as exponentials or as rational Cayley transforms are
considered here. Structural features of the results are discussed and compared,
and large j limits of the coefficients are examined. | math-ph |
Riemann-Hilbert approach and N-soliton solution for an eighth-order
nonlinear Schrodinger equation in an optical fiber: This paper aims to present an application of Riemann-Hilbert approach to
treat higher-order nonlinear differential equation that is an eighth-order
nonlinear Schrodinger equation arising in an optical fiber. Starting from the
spectral analysis of the Lax pair, a Riemann-Hilbert problem is formulated.
Then by solving the obtained Riemann-Hilbert problem under the reflectionless
case, N-soliton solution is generated for the eighth-order nonlinear
Schrodinger equation. Finally, the three-dimensional plots and two-dimensional
curves with specific choices of the involved parameters are made to show the
localized structures and dynamic behaviors of one- and two-soliton solutions. | math-ph |
On the WDVV equations in five-dimensional gauge theories: It is well-known that the perturbative prepotentials of four-dimensional N=2
supersymmetric Yang-Mills theories satisfy the generalized WDVV equations,
regardless of the gauge group. In this paper we study perturbative
prepotentials of the five-dimensional theories for some classical gauge groups
and determine whether or not they satisfy the WDVV system. | math-ph |
Parametric Cutoffs for Interacting Fermi Liquids: We introduce a new multiscale decomposition of the Fermi propagator based on
its parametric representation. We prove that the corresponding sliced
propagator obeys the same direct space bounds than the previous decomposition
used by the authors. Therefore non perturbative bounds on completely convergent
contributions still hold. In addition the new slicing better preserves momenta,
hence should become an important new technical tool for the rigorous analysis
of condensed matter systems. In particular it should allow to complete the
proof that a three dimensional interacting system of Fermions with spherical
Fermi surface is a Fermi liquid in the sense of Salmhofer's criterion. | math-ph |
Biorthogonal vectors, sesquilinear forms and some physical operators: Continuing the analysis undertaken in previous articles, we discuss some
features of non-self-adjoint operators and sesquilinear forms which are defined
starting from two biorthogonal families of vectors, like the so-called
generalized Riesz systems, enjoying certain properties. In particular we
discuss what happens when they forms two $\D$-quasi bases. | math-ph |
Density and current profiles in $U_q(A^{(1)}_2)$ zero range process: The stochastic $R$ matrix for $U_q(A^{(1)}_n)$ introduced recently gives rise
to an integrable zero range process of $n$ classes of particles in one
dimension. For $n=2$ we investigate how finitely many first class particles
fixed as defects influence the grand canonical ensemble of the second class
particles. By using the matrix product stationary probabilities involving
infinite products of $q$-bosons, exact formulas are derived for the local
density and current of the second class particles in the large volume limit. | math-ph |
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