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Vertices from replica in a random matrix theory: Kontsevitch's work on Airy matrix integrals has led to explicit results for
the intersection numbers of the moduli space of curves. In a subsequent work
Okounkov rederived these results from the edge behavior of a Gaussian matrix
integral. In our work we consider the correlation functions of vertices in a
Gaussian random matrix theory, with an external matrix source, in a scaling
limit in which the powers of the matrices and their sizes go to infinity
simultaneously in a specified scale. We show that the replica method applied to
characteristic polynomials of the random matrices, together with a duality
exchanging N and the number of points, allows one to recover Kontsevich's
results on the intersection numbers, through a simple saddle-point analysis. | math-ph |
Universal K-matrix distribution in beta=2 Ensembles of Random Matrices: The K-matrix, also known as the "Wigner reaction matrix" in nuclear
scattering or "impedance matrix" in the electromagnetic wave scattering, is
given essentially by an M x M diagonal block of the resolvent (E-H)^{-1} of a
Hamiltonian H. For chaotic quantum systems the Hamiltonian H can be modelled by
random Hermitian N x N matrices taken from invariant ensembles with the Dyson
symmetry index beta=1,2,4. For beta=2 we prove by explicit calculation a
universality conjecture by P. Brouwer which is equivalent to the claim that the
probability distribution of K, for a broad class of invariant ensembles of
random Hermitian matrices H, converges to a matrix Cauchy distribution with
density ${\cal P}(K)\propto
\left[\det{({\lambda}^2+(K-{\epsilon})^2)}\right]^{-M}$ in the limit $N\to
\infty$, provided the parameter M is fixed and the spectral parameter E is
taken within the support of the eigenvalue distribution of H. In particular, we
show that for a broad class of unitary invariant ensembles of random matrices
finite diagonal blocks of the resolvent are Cauchy distributed. The cases
beta=1 and beta=4 remain outstanding. | math-ph |
Resonances and inverse problems for energy-dependent potentials on the
half-line: We consider Schr\"{o}dinger equations with linearly energy-depending
potentials which are compactly supported on the half-line. We first provide
estimates of the number of eigenvalues and resonances for such complex-valued
potentials under suitable regularity assumptions. Then, we consider a specific
class of energy-dependent Schr\"{o}dinger equations without eigenvalues,
defined with Miura potentials and boundary conditions at the origin. We solve
the inverse resonance problem in this case and describe sets of iso-resonance
potentials and boundary condition parameters. Our strategy consists in
exploiting a correspondance between Schr\"{o}dinger and Dirac equations on the
half-line. As a byproduct, we describe similar sets for Dirac operators and
show that the scattering problem for Schr\"{o}dinger equation or Dirac operator
with an arbitrary boundary condition can be reduced to the scattering problem
with the Dirichlet boundary condition. | math-ph |
Fractional Lattice Dynamics: Nonlocal constitutive behavior generated by
power law matrix functions and their fractional continuum limit kernels: We introduce positive elastic potentials in the harmonic approximation
leading by Hamilton's variational principle to fractional Laplacian matrices
having the forms of power law matrix functions of the simple local Bornvon
Karman Laplacian. The fractional Laplacian matrices are well defined on
periodic and infinite lattices in $n=1,2,3,..$ dimensions. The present approach
generalizes the central symmetric second differenceoperator (Born von Karman
Laplacian) to its fractional central symmetric counterpart (Fractional
Laplacian matrix).For non-integer powers of the Born von Karman Laplacian, the
fractional Laplacian matrix is nondiagonal with nonzero matrix elements
everywhere, corresponding to nonlocal behavior: For large lattices the matrix
elements far from the diagonal expose power law asymptotics leading to
continuum limit kernels of Riesz fractional derivative type. We present
explicit results for the fractional Laplacian matrix in 1D for finite periodic
and infinite linear chains and their Riesz fractional derivative continuum
limit kernels.The approach recovers for $\alpha=2$ the well known classical
Born von Karman linear chain (1D lattice) with local next neighbor
springsleading in the well known continuum limit of classic local standard
elasticity, and for other integer powers to gradient elasticity.We also present
a generalization of the fractional Laplacian matrix to n-dimensional cubic
periodic (nD tori) and infinite lattices. For the infinite nD lattice we
deducea convenient integral representation.We demonstrate that our fractional
lattice approach is a powerful tool to generate physically admissible nonlocal
lattice material models and their continuum representations. | math-ph |
Perturbative calculation of energy levels for the Dirac equation with
generalised momenta: We analyse a modified Dirac equation based on a noncommutative structure in
phase space. The noncommutative structure induces generalised momenta and
contributions to the energy levels of the standard Dirac equation. Using
techniques of perturbation theory, we use this approach to find the lowest
order corrections to the energy levels and eigenfunctions for two linear
potentials in three dimensions, one with radial dependence and another with a
triangular shape along one spatial dimension. We find that the corrections due
to the noncommutative contributions may be of the same order as the
relativistic ones. | math-ph |
The incipient infinite cluster in high-dimensional percolation: We announce our recent proof that, for independent bond percolation in high
dimensions, the scaling limits of the incipient infinite cluster's two-point
and three-point functions are those of integrated super-Brownian excursion
(ISE). The proof uses an extension of the lace expansion for percolation. | math-ph |
Random Möbius dynamics on the unit disc and perturbation theory for
Lyapunov exponents: Randomly drawn $2\times 2$ matrices induce a random dynamics on the Riemann
sphere via the M\"obius transformation. Considering a situation where this
dynamics is restricted to the unit disc and given by a random rotation
perturbed by further random terms depending on two competing small parameters,
the invariant (Furstenberg) measure of the random dynamical system is
determined. The results have applications to the perturbation theory of
Lyapunov exponents which are of relevance for one-dimensional discrete random
Schr\"odinger operators. | math-ph |
Cosmic strings in a generalized linear formulation of gauge field theory: In this note we construct self-dual cosmic strings from a gauge field theory
with a generalized linear formation of potential energy density. By integrating
the Einstein equation, we obtain a nonlinear elliptic equation which is equal
with the sources. We prove the existence of a solution in the broken symmetry
category on the full plane and the multiple string solutions are valid under a
sufficient condition imposed only on the total string number N. The technique
of upper-lower solutions and the method of regularization are employed to show
the existence of a solution when there are at least two distant string centers.
When all the string centers are identical, fixed point theorem are used to
study the properties of the nonlinear elliptic equation. Finally, We give the
sharp asymptotic estimate for the solution at infinity. | math-ph |
Partial Reductions of Hamiltonian Flows and Hess-Appel'rot Systems on
SO(n): We study reductions of the Hamiltonian flows restricted to their invariant
submanifolds. As examples, we consider partial Lagrange-Routh reductions of the
natural mechanical systems such as geodesic flows on compact Lie groups and
$n$-dimensional variants of the classical Hess-Appel'rot case of a heavy rigid
body motion about a fixed point. | math-ph |
On the covariant Hamilton-Jacobi formulation of Maxwell's equations via
the polysymplectic reduction: The covariant Hamilton-Jacobi formulation of Maxwell's equations is derived
from the first-order (Palatini-like) Lagrangian using the analysis of
constraints within the De~Donder-Weyl covariant Hamiltonian formalism and the
corresponding polysymplectic reduction. | math-ph |
An example of double confluent Heun equation: Schroedinger equation with
supersingular plus Coulomb potential: A recently proposed algorithm to obtain global solutions of the double
confluent Heun equation is applied to solve the quantum mechanical problem of
finding the energies and wave functions of a particle bound in a potential sum
of a repulsive supersingular term, Ar(-4), plus an attractive Coulombian one,
-Zr(-1). The existence of exact algebraic solutions for certain values of A is
discussed. | math-ph |
Born-Oppenheimer potential energy surfaces for Kohn-Sham models in the
local density approximation: We show that the Born-Oppenheimer potential energy surface in Kohn-Sham
theory behaves like the corresponding one in Thomas-Fermi theory up to
$o(R^{-7})$ for small nuclear separation $R$. We also prove that if a
minimizing configuration exists, then the minimal distance of nuclei is larger
than some constant which is independent of the nuclear charges. | math-ph |
Stability of atoms and molecules in an ultrarelativistic
Thomas-Fermi-Weizsaecker model: We consider the zero mass limit of a relativistic Thomas-Fermi-Weizsaecker
model of atoms and molecules. We find bounds for the critical nuclear charges
that ensure stability. | math-ph |
Painlevé transcendent evaluations of finite system density matrices
for 1d impenetrable Bosons: The recent experimental realisation of a one-dimensional Bose gas of ultra
cold alkali atoms has renewed attention on the theoretical properties of the
impenetrable Bose gas. Of primary concern is the ground state occupation of
effective single particle states in the finite system, and thus the tendency
for Bose-Einstein condensation. This requires the computation of the density
matrix. For the impenetrable Bose gas on a circle we evaluate the density
matrix in terms of a particular Painlev\'e VI transcendent in $\sigma$-form,
and furthermore show that the density matrix satisfies a recurrence relation in
the number of particles. For the impenetrable Bose gas in a harmonic trap, and
with Dirichlet or Neumann boundary conditions, we give a determinant form for
the density matrix, a form as an average over the eigenvalues of an ensemble of
random matrices, and in special cases an evaluation in terms of a transcendent
related to Painlev\'e V and VI. We discuss how our results can be used to
compute the ground state occupations. | math-ph |
Infinitely many shape invariant potentials and new orthogonal
polynomials: Three sets of exactly solvable one-dimensional quantum mechanical potentials
are presented. These are shape invariant potentials obtained by deforming the
radial oscillator and the trigonometric/hyperbolic P\"oschl-Teller potentials
in terms of their degree \ell polynomial eigenfunctions. We present the entire
eigenfunctions for these Hamiltonians (\ell=1,2,...) in terms of new orthogonal
polynomials. Two recently reported shape invariant potentials of Quesne and
G\'omez-Ullate et al's are the first members of these infinitely many
potentials. | math-ph |
A unified approach to Hamiltonian systems, Poisson systems, gradient
systems, and systems with Lyapunov functions and/or first integrals: Systems with a first integral (i.e., constant of motion) or a Lyapunov
function can be written as ``linear-gradient systems'' $\dot x= L(x)\nabla
V(x)$ for an appropriate matrix function $L$, with a generalization to several
integrals or Lyapunov functions. The discrete-time analogue, $\Delta x/\Delta t
= L \bar\nabla V$ where $\bar\nabla$ is a ``discrete gradient,'' preserves $V$
as an integral or Lyapunov function, respectively. | math-ph |
Hecke Transformations of Conformal Blocks in WZW Theory. I. KZB
Equations for Non-Trivial Bundles: We describe new families of the Knizhnik-Zamolodchikov-Bernard (KZB)
equations related to the WZW-theory corresponding to the adjoint $G$-bundles of
different topological types over complex curves $\Sigma_{g,n}$ of genus $g$
with $n$ marked points. The bundles are defined by their characteristic classes
- elements of $H^2(\Sigma_{g,n},\mathcal{Z}(G))$, where $\mathcal{Z}(G)$ is a
center of the simple complex Lie group $G$. The KZB equations are the
horizontality condition for the projectively flat connection (the KZB
connection) defined on the bundle of conformal blocks over the moduli space of
curves. The space of conformal blocks has been known to be decomposed into a
few sectors corresponding to the characteristic classes of the underlying
bundles. The KZB connection preserves these sectors. In this paper we construct
the connection explicitly for elliptic curves with marked points and prove its
flatness. | math-ph |
Structures Preserved by Consistently Graded Lie Superalgebras: Dual Pfaff equations (of the form \tilde D^a = 0, \tilde D^a some vector
fields of degree -1) preserved by the exceptional infinite-dimensional simple
Lie superalgebras ksle(5|10), vle(3|6) and mb(3|8) are constructed, yielding an
intrinsic geometric definition of these algebras. This leads to conditions on
the vector fields, which are solved explicitly. Expressions for preserved
differential form equations (Pfaff equations), brackets (similar to contact
brackets) and tensor modules are written down. The analogous construction for
the contact superalgebra k(1|m) (a.k.a. the centerless N=m superconformal
algebra) is reviewed. | math-ph |
Phase-field gradient theory: We propose a phase-field theory for enriched continua. To generalize
classical phase-field models, we derive the phase-field gradient theory based
on balances of microforces, microtorques, and mass. We focus on materials where
second gradients of the phase field describe long-range interactions. By
considering a nontrivial interaction inside the body, described by a
boundary-edge microtraction, we characterize the existence of a
microhypertraction field, a central aspect of this theory. On surfaces, we
define the surface microtraction and the surface-couple microtraction emerging
from internal surface interactions. We explicitly account for the lack of
smoothness along a curve on surfaces enclosing arbitrary parts of the domain.
In these rough areas, internal-edge microtractions appear. We begin our theory
by characterizing these tractions. Next, in balancing microforces and
microtorques, we arrive at the field equations. Subject to thermodynamic
constraints, we develop a general set of constitutive relations for a
phase-field model where its free-energy density depends on second gradients of
the phase field. A priori, the balance equations are general and independent of
constitutive equations, where the thermodynamics constrain the constitutive
relations through the free-energy imbalance. To exemplify the usefulness of our
theory, we generalize two commonly used phase-field equations. We propose a
'generalized Swift-Hohenberg equation'-a second-grade phase-field equation-and
its conserved version, the 'generalized phase-field crystal equation'-a
conserved second-grade phase-field equation. Furthermore, we derive the
configurational fields arising in this theory. We conclude with the
presentation of a comprehensive, thermodynamically consistent set of boundary
conditions. | math-ph |
Expression of the Holtsmark function in terms of hypergeometric $_2F_2$
and Airy $\mathrm{Bi}$ functions: The Holtsmark distribution has applications in plasma physics, for the
electric-microfield distribution involved in spectral line shapes for instance,
as well as in astrophysics for the distribution of gravitating bodies. It is
one of the few examples of a stable distribution for which a closed-form
expression of the probability density function is known. However, the latter is
not expressible in terms of elementary functions. In the present work, we
mention that the Holtsmark probability density function can be expressed in
terms of hypergeometric function $_2F_2$ and of Airy function of the second
kind $\mathrm{Bi}$ and its derivative. The new formula is simpler than the one
proposed by Lee involving $_2F_3$ and $_3F_4$ hypergeometric functions. | math-ph |
Symmetry properties of Penrose type tilings: The Penrose tiling is directly related to the atomic structure of certain
decagonal quasicrystals and, despite its aperiodicity, is highly symmetric. It
is known that the numbers 1, $-\tau $, $(-\tau)^2$, $(-\tau)^3$, ..., where
$\tau =(1+\sqrt{5})/2$, are scaling factors of the Penrose tiling. We show that
the set of scaling factors is much larger, and for most of them the number of
the corresponding inflation centers is infinite. | math-ph |
Ground states of Nicolai and $\mathbb{Z}_2$ Nicolai models: We derive explicit recursions for the ground state generating functions of
the one-dimensional Nicolai model and $\mathbb{Z}_2$ Nicolai model. Both are
examples of lattice models with $\mathcal{N}=2$ supersymmetry. The relations
that we obtain for the $\mathbb{Z}_2$ model were numerically predicted by
Sannomiya, Katsura, and Nakayama. | math-ph |
Anderson's orthogonality catastrophe in one dimension induced by a
magnetic field: According to Anderson's orthogonality catastrophe, the overlap of the
$N$-particle ground states of a free Fermi gas with and without an (electric)
potential decays in the thermodynamic limit. For the finite one-dimensional
system various boundary conditions are employed. Unlike the usual setup the
perturbation is introduced by a magnetic (vector) potential. Although such a
magnetic field can be gauged away in one spatial dimension there is a
significant and interesting effect on the overlap caused by the phases. We
study the leading asymptotics of the overlap of the two ground states and the
two-term asymptotics of the difference of the ground-state energies. In the
case of periodic boundary conditions our main result on the overlap is based
upon a well-known asymptotic expansion by Fisher and Hartwig on Toeplitz
determinants with a discontinuous symbol. In the case of Dirichlet boundary
conditions no such result is known to us and we only provide an upper bound on
the overlap, presumably of the right asymptotic order. | math-ph |
Differential calculus and connections on a quantum plane at a cubic root
of unity: We consider the algebra of N x N matrices as a reduced quantum plane on which
a finite-dimensional quantum group H acts. This quantum group is a quotient of
U_q(sl(2,C)), q being an N-th root of unity. Most of the time we shall take
N=3; in that case \dim(H) = 27. We recall the properties of this action and
introduce a differential calculus for this algebra: it is a quotient of the
Wess-Zumino complex. The quantum group H also acts on the corresponding
differential algebra and we study its decomposition in terms of the
representation theory of H. We also investigate the properties of connections,
in the sense of non commutative geometry, that are taken as 1-forms belonging
to this differential algebra. By tensoring this differential calculus with
usual forms over space-time, one can construct generalized connections with
covariance properties with respect to the usual Lorentz group and with respect
to a finite-dimensional quantum group. | math-ph |
Axiomatic field theory and Hida-Colombeau algebras: An axiomatic quantum field theory applied to the self-interacting boson field
is realised in terms of generalised operators that allows us to form products
and take derivatives of the fields in simple and mathematically rigorous ways.
Various spaces are explored for representation of these operators with this
exploration culminating with a Hida-Colombeau algebra. Rigorous well defned
Hamiltonians are written using ordinary products of interacting scalar fields
that are represented as generalised operators on simplified Hida-Colombeau
algebras. | math-ph |
Exponential Mixing of the 2D Stochastic Navier-Stokes Dynamics: We consider the Navier-Stokes equation on a two dimensional torus with a
random force which is white noise in time, and excites only a finite number of
modes. The number of excited modes depends on the viscosity $\nu$, and grows
like $\nu^{-3}$ when $\nu$ goes to zero. We prove that this Markov process has
a unique invariant measure and is exponentially mixing in time. | math-ph |
Integration over connections in the discretized gravitational functional
integrals: The result of performing integrations over connection type variables in the
path integral for the discrete field theory may be poorly defined in the case
of non-compact gauge group with the Haar measure exponentially growing in some
directions. This point is studied in the case of the discrete form of the first
order formulation of the Einstein gravity theory. Here the result of interest
can be defined as generalized function (of the rest of variables of the type of
tetrad or elementary areas) i. e. a functional on a set of probe functions. To
define this functional, we calculate its values on the products of components
of the area tensors, the so-called moments. The resulting distribution (in
fact, probability distribution) has singular ($\delta$-function-like) part with
support in the nonphysical region of the complex plane of area tensors and
regular part (usual function) which decays exponentially at large areas. As we
discuss, this also provides suppression of large edge lengths which is
important for internal consistency, if one asks whether gravity on short
distances can be discrete. Some another features of the obtained probability
distribution including occurrence of the local maxima at a number of the
approximately equidistant values of area are also considered. | math-ph |
Symmetries and geometrically implied nonlinearities in mechanics and
field theory: Discussed is relationship between nonlinearity and symmetry of dynamical
models. The special stress is laid on essential, non-perturbative nonlinearity,
when none linear background does exist. This is nonlinearity essentially
different from ones given by nonlinear corrections imposed onto some linear
background. In a sense our ideas follow and develop those underlying
Born-Infeld electrodynamics and general relativity. We are particularly
interested in affine symmetry of degrees of freedom and dynamical models.
Discussed are mechanical geodetic models where the elastic dynamics of the body
is not encoded in potential energy but rather in affinely-invariant kinetic
energy, i.e., in affinely-invariant metric tensors on the configuration space.
In a sense this resembles the idea of Maupertuis variational principle. We
discuss also the dynamics of the field of linear frames, invariant under the
action of linear group of internal symmetries. It turns out that such models
have automatically the generalized Born-Infeld structure. This is some new
justification of Born-Infeld ideas. The suggested models may be applied in
nonlinear elasticity and in mechanics of relativistic continua with
microstructure. They provide also some alternative models of gravitation
theory. There exists also some interesting relationship with the theory of
nonlinear integrable lattices. | math-ph |
Pseudo-bosons for the $D_2$ type quantum Calogero model: In the first part of this paper we show how a simple system, a 2-dimensional
quantum harmonic oscillator, can be described in terms of pseudo-bosonic
variables. This apparently {\em strange} choice is useful when the {\em
natural} Hilbert space of the system, $L^2({\bf R}^2)$ in this case, is, for
some reason, not the most appropriate. This is exactly what happens for the
$D_2$ type quantum Calogero model considered in the second part of the paper,
where the Hilbert space $L^2({\bf R}^2)$ appears to be an unappropriate choice,
since the eigenvectors of the relevant hamiltonian are not square-integrable.
Then we discuss how a certain intertwining operator arising from the model can
be used to fix a different Hilbert space more {\em useful}. | math-ph |
Gibbs measures for SOS models with external field on a Cayley tree: We consider a nearest-neighbor solid-on-solid (SOS) model, with several spin
values $0,1,\ldots,m,$ $m\geq2,$ and nonzero external field, on a Cayley tree
of degree $k$ (with $k+1$ neighbors). We are aiming to extend the results of
\cite{rs} where the SOS model is studied with (mainly) three spin values and
zero external field. The SOS model can be treated as a natural generalization
of the Ising model (obtained for $m=1$). We mainly assume that $m=2$ (three
spin values) and study translation-invariant (TI) and splitting (S) Gibbs
measures (GMs). (Splitting GMs have a particular Markov-type property specific
for a tree.) For $m=2$, in the antiferromagnet (AFM) case, a TISGM is unique
for all temperatures with an external field. In the ferromagnetic (FM) case,
for $m=2,$ the number of TISGMs varies with the temperature and the external
field: this gives an interesting example of phase transition.
Our second result gives a classification of all TISGMs of the Three-State
SOS-Model on the Cayley tree of degree two with the presence of an external
field. We show uniqueness in the case of antiferromagnetic interactions and the
existence of up to seven TISGMs in the case of ferromagnetic interactions,
where the number of phases depends on the interaction strength and external
field. | math-ph |
Orbit determination for standard-like maps: asymptotic expansion of the
confidence region in regular zones: We deal with the orbit determination problem for a class of maps of the
cylinder generalizing the Chirikov standard map. The problem consists of
determining the initial conditions and other parameters of an orbit from some
observations. A solution to this problem goes back to Gauss and leads to the
least squares method. Since the observations admit errors, the solution comes
with a confidence region describing the uncertainty of the solution itself. We
study the behavior of the confidence region in the case of a simultaneous
increase of the number of observations and the time span over which they are
performed. More precisely, we describe the geometry of the confidence region
for solutions in regular zones. We prove an estimate of the trend of the
uncertainties in a set of positive measure of the phase space, made of
invariant curve. Our result gives an analytical proof of some known numerical
evidences. | math-ph |
Operator-GENERIC Formulation of Thermodynamics of Irreversible Processes: Metriplectic systems are state space formulations that have become well-known
under the acronym GENERIC. In this work we present a GENERIC based state space
formulation in an operator setting that encodes a weak-formulation of the field
equations describing the dynamics of a homogeneous mixture of compressible
heat-conducting Newtonian fluids consisting of reactive constituents. We
discuss the mathematical model of the fluid mixture formulated in the framework
of continuum thermodynamics. The fluid mixture is considered an open
thermodynamic system that moves free of external body forces. As closure
relations we use the linear constitutive equations of the phenomenological
theory known as Thermodynamics of Irreversible Processes (TIP). The
phenomenological coefficients of these linear constitutive equations satisfy
the Onsager-Casimir reciprocal relations. We present the state space
representation of the fluid mixture, formulated in the extended GENERIC
framework for open systems, specified by a symmetric, mixture related
dissipation bracket and a mixture related Poisson-bracket for which we prove
the Jacobi-identity. | math-ph |
Quantum Electrodynamics of Atomic Resonances: A simple model of an atom interacting with the quantized electromagnetic
field is studied. The atom has a finite mass $m$, finitely many excited states
and an electric dipole moment, $\vec{d}_0 = -\lambda_{0} \vec{d}$, where $\|
d^{i}\| = 1,$ $ i=1,2,3,$ and $\lambda_0$ is proportional to the elementary
electric charge. The interaction of the atom with the radiation field is
described with the help of the Ritz Hamiltonian, $-\vec{d}_0\cdot \vec{E}$,
where $\vec{E}$ is the electric field, cut off at large frequencies. A
mathematical study of the Lamb shift, the decay channels and the life times of
the excited states of the atom is presented. It is rigorously proven that these
quantities are analytic functions of the momentum $\vec{p}$ of the atom and of
the coupling constant $\lambda_0$, provided $|\vec{p}| < mc$ and $| \Im\vec{p}
|$ and $| \lambda_{0} |$ are sufficiently small. The proof relies on a somewhat
novel inductive construction involving a sequence of `smooth Feshbach-Schur
maps' applied to a complex dilatation of the original Hamiltonian, which yields
an algorithm for the calculation of resonance energies that converges
super-exponentially fast. | math-ph |
Geometry of Mechanics: We study the geometry underlying mechanics and its application to describe
autonomous and nonautonomous conservative dynamical systems of different types;
as well as dissipative dynamical systems. We use different geometric
descriptions to study the main properties and characteristics of these systems;
such as their Lagrangian, Hamiltonian and unified formalisms, their symmetries,
the variational principles, and others. The study is done mainly for the
regular case, although some comments and explanations about singular systems
are also included. | math-ph |
Finding and solving Calogero-Moser type systems using Yang-Mills gauge
theories: Yang-Mills gauge theory models on a cylinder coupled to external matter
charges provide powerful means to find and solve certain non-linear integrable
systems. We show that, depending on the choice of gauge group and matter
charges, such a Yang-Mills model is equivalent to trigonometric Calogero-Moser
systems and certain known spin generalizations thereof. Choosing a more general
ansatz for the matter charges allows us to obtain and solve novel integrable
systems. The key property we use to prove integrability and to solve these
systems is gauge invariance of the corresponding Yang-Mills model. | math-ph |
Constraints in polysymplectic (covariant) Hamiltonian formalism: In the framework of polysymplectic Hamiltonian formalism, degenerate
Lagrangian field systems are described as multi-Hamiltonian systems with
Lagrangian constraints. The physically relevant case of degenerate quadratic
Lagrangians is analized in detail, and the Koszul--Tate resolution of
Lagrangian constraints is constructed in an explicit form. The particular case
of Hamiltonian mechanics with time-dependent constraints is studied. | math-ph |
Spacetime and observer space symmetries in the language of Cartan
geometry: We introduce a definition of symmetry generating vector fields on manifolds
which are equipped with a first-order reductive Cartan geometry. We apply this
definition to a number of physically motivated examples and show that our newly
introduced notion of symmetry agrees with the usual notions of symmetry of
affine, Riemann-Cartan, Riemannian and Weizenb\"ock geometries, which are
conventionally used as spacetime models. Further, we discuss the case of Cartan
geometries which can be used to model observer space instead of spacetime. We
show which vector fields on an observer space can be interpreted as symmetry
generators of an underlying spacetime manifold, and may hence be called
"spatio-temporal". We finally apply this construction to Finsler spacetimes and
show that symmetry generating vector fields on a Finsler spacetime are indeed
in a one-to-one correspondence with spatio-temporal vector fields on its
observer space. | math-ph |
On consistency of perturbed generalised minimal models: We consider the massive {perturbation} of the Generalised Minimal Model
introduced by Al. Zamolodchikov. The one-point functions in this case are
supposed to be described by certain function $\omega(\z,\xi)$. We prove that
this function satisfies several properties which are necessary for consistency
of the entire procedure. | math-ph |
Relativistic Entropy Inequality: In this paper we apply the entropy principle to the relativistic version of
the differential equations describing a standard fluid flow, that is, the
equations for mass, momentum, and a system for the energy matrix. These are the
second order equations which have been introduced in [3]. Since the principle
also says that the entropy equation is a scalar equation, this implies, as we
show, that one has to take a trace in the energy part of the system. Thus one
arrives at the relativistic mass-momentum-energy system for the fluid. In the
procedure we use the well-known Liu-M\"uller sum [10] in order to deduce the
Gibbs relation and the residual entropy inequality. | math-ph |
Dynamics of spiral waves in the complex Ginzburg-Landau equation in
bounded domains: Multiple-spiral-wave solutions of the general cubic complex Ginzburg-Landau
equation in bounded domains are considered. We investigate the effect of the
boundaries on spiral motion under homogeneous Neumann boundary conditions, for
small values of the twist parameter $q$. We derive explicit laws of motion for
rectangular domains and we show that the motion of spirals becomes
exponentially slow when the twist parameter exceeds a critical value depending
on the size of the domain. The oscillation frequency of multiple-spiral
patterns is also analytically obtained. | math-ph |
Symplectic Coarse-Grained Dynamics: Chalkboard Motion in Classical and
Quantum Mechanics: In the usual approaches to mechanics (classical or quantum) the primary
object of interest is the Hamiltonian, from which one tries to deduce the
solutions of the equations of motion (Hamilton or Schr\"odinger). In the
present work we reverse this paradigm and view the motions themselves as being
the primary objects. This is made possible by studying arbitrary phase space
motions, not of points, but of (small) ellipsoids with the requirement that the
symplectic capacity of these ellipsoids is preserved. This allows us to guide
and control these motions as we like. In the classical case these ellipsoids
correspond to a symplectic coarse graining of phase space, and in the quantum
case they correspond to the "quantum blobs" we defined in previous work, and
which can be viewed as minimum uncertainty phase space cells which are in a
one-to-one correspondence with Gaussian pure states. | math-ph |
An inclusive curvature-like framework for describing dissipation:
metriplectic 4-bracket dynamics: An inclusive framework for joined Hamiltonian and dissipative dynamical
systems, which preserve energy and produce entropy, is given. The dissipative
dynamics of the framework is based on the metriplectic 4-bracket, a quantity
like the Poisson bracket defined on phase space functions, but unlike the
Poisson bracket has four slots with symmetries and properties motivated by
Riemannian curvature. Metriplectic 4-bracket dynamics is generated using two
generators, the Hamiltonian and the entropy, with the entropy being a Casimir
of the Hamiltonian part of the system. The formalism includes all known
previous binary bracket theories for dissipation or relaxation as special
cases. Rich geometrical significance of the formalism and methods for
constructing metriplectic 4-brackets are explored. Many examples of both finite
and infinite dimensions are given. | math-ph |
For the Quantum Heisenberg Ferromagnet, a Polymer Expansion and its High
T Convergence: We let Psi_0 be a wave function for the Quantum Heisenberg ferromagnet sharp
i sigma_zi and Psi_mu = exp(-mu*H)Psi_0. We study expectations similar to the
form <Psi_mu,(sigma_zi)Psi_mu>/<Psi_mu,Psi_mu> for which we present a formal
polymer expansion, whose convergence we prove for sufficiently small mu.
The approach of the paper is to relate the wavefunction Psi_mu to an
approximation to it that is a product function. In the jth spot of the product
approximation the upper component is phi_mu(j), and the lower component is
(1-phi_mu(j)), where phi satisfies the lattice heat equation. This is shown via
a cluster or polymer expansion.
The present work began in a previous paper, primarily a numerical study, and
provides a proof of results related to Conjecture 3 of this previous paper. | math-ph |
Symmetry of Lie algebras associated with (ε,δ)
Freudenthal-Kantor triple systems: Symmetry group of Lie algebras and superalgebras constructed from
(\epsilon,\delta) Freudenthal- Kantor triple systems has been studied.
Especially, for a special (\epsilon,\epsilon) Freudenthal- Kantor triple, it is
SL(2) group. Also, relationship between two constructions of Lie algebras from
structurable algebra has been investigated. | math-ph |
The Riemann-Hilbert approach to double scaling limit of random matrix
eigenvalues near the "birth of a cut" transition: In this paper we studied the double scaling limit of a random unitary matrix
ensemble near a singular point where a new cut is emerging from the support of
the equilibrium measure. We obtained the asymptotic of the correlation kernel
by using the Riemann-Hilbert approach. We have shown that the kernel near the
critical point is given by the correlation kernel of a random unitary matrix
ensemble with weight $e^{-x^{2\nu}}$. This provides a rigorous proof of the
previous results of Eynard. | math-ph |
Uniform N-particle Anderson localization and unimodal eigenstates in
deterministic disordered media without induction on the number of particles: We present the first rigorous result on Anderson localization for interacting
systems of quantum particles subject to a deterministic (e.g., almost periodic)
disordered external potential. For a particular class of deterministic,
fermionic, Anderson-type Hamiltonians on the lattice of an arbitrary dimension,
and for a large class of underlying dynamical systems generating the external
potential, we prove that the spectrum is pure point, all eigenstates are
unimodal and feature a uniform exponential decay. In contrast to all prior
mathematical works on multi-particle Anderson localization, we do not use the
induction on the number of particles. | math-ph |
Two-Parameter Dynamics and Geometry: In this paper we present the two-parameter dynamics which is implied by the
law of inertia in flat spacetime. A remarkable perception is that (A)dS4
geometry may emerge from the two-parameter dynamics, which exhibits some
phenomenon of dynamics/ geometry correspondence. We also discuss the Unruh
effects within the context of two-parameter dynamics. In the last section we
construct various invariant actions with respect to the broken symmetry groups. | math-ph |
Matrix continued fraction solution to the relativistic spin-$0$
Feshbach-Villars equations: The Feshbach-Villars equations, like the Klein-Gordon equation, are
relativistic quantum mechanical equations for spin-$0$ particles. We write the
Feshbach-Villars equations into an integral equation form and solve them by
applying the Coulomb-Sturmian potential separable expansion method. We consider
bound-state problems in a Coulomb plus short range potential. The corresponding
Feshbach-Villars Coulomb Green's operator is represented by a matrix continued
fraction. | math-ph |
Liouville-type equations for the n-particle distribution functions of an
open system: In this work we derive a mathematical model for an open system that exchanges
particles and momentum with a reservoir from their joint Hamiltonian dynamics.
The complexity of this many-particle problem is addressed by introducing a
countable set of n-particle phase space distribution functions just for the
open subsystem, while accounting for the reservoir only in terms of statistical
expectations. From the Liouville equation for the full system we derive a set
of coupled Liouville-type equations for the n-particle distributions by
marginalization with respect to reservoir states. The resulting equation
hierarchy describes the external momentum forcing of the open system by the
reservoir across its boundaries, and it covers the effects of particle
exchanges, which induce probability transfers between the n- and (n+1)-particle
distributions. Similarities and differences with the Bergmann-Lebowitz model of
open systems (P.G.Bergmann, J.L. Lebowitz, Phys.Rev., 99:578--587 (1955)) are
discussed in the context of the implementation of these guiding principles in a
computational scheme for molecular simulations. | math-ph |
The $κ$-(A)dS noncommutative spacetime: The (3+1)-dimensional $\kappa$-(A)dS noncommutative spacetime is explicitly
constructed by quantizing its semiclassical counterpart, which is the
$\kappa$-(A)dS Poisson homogeneous space. This turns out to be the only
possible generalization of the well-known $\kappa$-Minkowski spacetime to the
case of non-vanishing cosmological constant, under the condition that the time
translation generator of the corresponding quantum (A)dS algebra is primitive.
Moreover, the $\kappa$-(A)dS noncommutative spacetime is shown to have a
quadratic subalgebra of local spatial coordinates whose first-order brackets in
terms of the cosmological constant parameter define a quantum sphere, while the
commutators between time and space coordinates preserve the same structure of
the $\kappa$-Minkowski spacetime. When expressed in ambient coordinates, the
quantum $\kappa$-(A)dS spacetime is shown to be defined as a noncommutative
pseudosphere. | math-ph |
The time-averaged limit measure of the Wojcik model: We investigate "the Wojcik model" introduced and studied by Wojcik et al.,
which is a one-defect quantum walk (QW) having a single phase at the origin.
They reported that giving a phase at one point causes an astonishing effect for
localization. There are three types of measures having important roles in the
study of QWs: time-averaged limit measure, weak limit measure, and stationary
measure. The first two measures imply a coexistence of localized behavior and
the ballistic spreading in the QW. As Konno et al. suggested, the time-averaged
limit and stationary measures are closely related to each other for some
models. In this paper, we focus on a relation between the two measures for the
Wojcik model. The stationary measure was already obtained by our previous work.
Here, we get the time-averaged limit measure by several methods. Our results
show that the stationary measure is a special case of the time-averaged limit
measure. | math-ph |
Infinite Dimensional Choi-Jamiolkowski States and Time Reversed Quantum
Markov Semigroups: We propose a definition of infinite dimensional Choi-Jamiolkowski state
associated with a completely positive trace preserving map. We introduce the
notion of Theta-KMS adjoint of a quantum Markov semigroup, which is identified
with the time reversed semigroup. The break down of Theta-KMS symmetry (or
Theta-standard quantum detailed balance in the sense of Fagnola-Umanita) is
measured by means of the von Neumann relative entropy of the Choi-Jamiolkowski
states associated with the semigroup and its Theta-KMS adjoint. | math-ph |
On Derivation of the Poisson-Boltzmann Equation: Starting from the microscopic reduced Hartree-Fock equation, we derive the
nanoscopic linearized Poisson-Boltzmann equation for the electrostatic
potential associated with the electron density. | math-ph |
Operator Ordering and Solution of Pseudo-Evolutionary Equations: The solution of pseudo initial value differential equations, either ordinary
or partial (including those of fractional nature), requires the development of
adequate analytical methods, complementing those well established in the
ordinary differential equation setting. A combination of techniques, involving
procedures of umbral and of operational nature, has been demonstrated to be a
very promising tool in order to approach within a unifying context
non-canonical evolution problems. This article covers the extension of this
approach to the solution of pseudo-evolutionary equations. We will comment on
the explicit formulation of the necessary techniques, which are based on
certain time- and operator ordering tools. We will in particular demonstrate
how Volterra-Neumann expansions, Feynman-Dyson series and other popular tools
can be profitably extended to obtain solutions of fractional differential
equations. We apply the method to a number of examples, in which fractional
calculus and a certain umbral image calculus play a role of central importance. | math-ph |
Green Functions For Wave Propagation on a 5D manifold and the Associated
Gauge Fields Generated by a Uniformly Moving Point Source: Gauge fields associated with the manifestly covariant dynamics of particles
in (3,1) spacetime are five-dimensional. We provide solutions of the classical
5D gauge field equations in both (4,1) and (3,2) flat spacetime metrics for the
simple example of a uniformly moving point source. Green functions for the 5D
field equations are obtained, which are consistent with the solutions for
uniform motion obtained directly from the field equations with free asymptotic
conditions. | math-ph |
Equivariant wave maps exterior to a ball: We consider the exterior Cauchy-Dirichlet problem for equivariant wave maps
from 3+1 dimensional Minkowski spacetime into the three-sphere. Using mixed
analytical and numerical methods we show that, for a given topological degree
of the map, all solutions starting from smooth finite energy initial data
converge to the unique static solution (harmonic map). The asymptotics of this
relaxation process is described in detail. We hope that our model will provide
an attractive mathematical setting for gaining insight into
dissipation-by-dispersion phenomena, in particular the soliton resolution
conjecture. | math-ph |
Stretching magnetic fields by dynamo plasmas in Riemannian knotted tubes: Recently Shukurov et al [Phys Rev E 72, 025302 (2008)], made use of
non-orthogonal curvilinear coordinate system on a dynamo Moebius strip flow, to
investigate the effect of stretching by a turbulent liquid sodium flow. In
plasma physics, Chui and Moffatt [Proc Roy Soc A 451,609,(1995)] (CM),
considered non-orthogonal coordinates to investigate knotted magnetic flux tube
Riemann metric. Here it is shown that, in the unstretching knotted tubes,
dynamo action cannot be supported. Turbulence there, is generated by suddenly
braking of torus rotation. Here, use of CM metric, shows that stretching of
magnetic knots, by ideal plasmas, may support dynamo action. Investigation on
the stretching in plasma dynamos, showed that in diffusive media [Phys Plasma
\textbf{15},122106,(2008)], unstretching unknotted tubes do not support fast
dynamo action. Non-orthogonal coordinates in flux tubes of non-constant
circular section, of positive growth rate, leads to tube shrinking to a
constant value. As tube shrinks, curvature grows enhancing dynamo action. | math-ph |
Energy-dependent correlations in the $S$-matrix of chaotic systems: The $M$-dimensional unitary matrix $S(E)$, which describes scattering of
waves, is a strongly fluctuating function of the energy for complex systems
such as ballistic cavities, whose geometry induces chaotic ray dynamics. Its
statistical behaviour can be expressed by means of correlation functions of the
kind $\left \langle S_{ij}(E+\epsilon)S^\dag_{pq}(E-\epsilon)\right\rangle$,
which have been much studied within the random matrix approach. In this work,
we consider correlations involving an arbitrary number of matrix elements and
express them as infinite series in $1/M$, whose coefficients are rational
functions of $\epsilon$. From a mathematical point of view, this may be seen as
a generalization of the Weingarten functions of circular ensembles. | math-ph |
Effective-Mass Klein-Gordon-Yukawa Problem for Bound and Scattering
States: Bound and scattering state solutions of the effective-mass Klein-Gordon
equation are obtained for the Yukawa potential with any angular momentum
$\ell$. Energy eigenvalues, normalized wave functions and scattering phase
shifts are calculated as well as for the constant mass case. Bound state
solutions of the Coulomb potential are also studied as a limiting case.
Analytical and numerical results are compared with the ones obtained before. | math-ph |
Upper bounds of spin-density wave energies in the homogeneous electron
gas: Studying the jellium model in the Hartree-Fock approximation, Overhauser has
shown that spin density waves (SDW) can lower the energy of the Fermi gas, but
it is still unknown if these SDW are actually relevant for the phase diagram.
In this paper, we give a more complete description of SDW states. We show that
a modification of the Overhauser ansatz explains the behavior of the jellium at
high density compatible with previous Hartree-Fock simulations. | math-ph |
The complex elliptic Ginibre ensemble at weak non-Hermiticity: bulk
spacing distributions: We show that the distribution of bulk spacings between pairs of adjacent
eigenvalue real parts of a random matrix drawn from the complex elliptic
Ginibre ensemble is asymptotically given by a generalization of the
Gaudin-Mehta distribution, in the limit of weak non-Hermiticity. The same
generalization is expressed in terms of an integro-differential Painlev\'e
function and it is shown that the generalized Gaudin-Mehta distribution
describes the crossover, with increasing degree of non-Hermiticity, from
Gaudin-Mehta nearest-neighbor bulk statistics in the Gaussian Unitary Ensemble
to Poisson gap statistics for eigenvalue real parts in the bulk of the Complex
Ginibre Ensemble. | math-ph |
A formulation of Noether's theorem for fractional classical fields: This paper presents a formulation of Noether's theorem for fractional
classical fields.
We extend the variational formulations for fractional discrete systems to
fractional field systems. By applying the variational principle to a fractional
action $S$, we obtain the fractional Euler-Lagrange equations of motion.
Considerations of the Noether's variational problem for discrete systems whose
action is invariant under gauge transformations will be extended to fractional
variational problems for classical fields. The conservation laws associated
with fractional classical fields are derived. As an example we present the
conservation laws for the fractional Dirac fields. | math-ph |
Integrable models and star structures: We consider the representations of Hopf algebras involved in some physical
models, namely, factorizable S-matrix models (FSM's), one-dimensional quantum
spin chains (QSC's) and statistical vertex models (SVM's). These physical
representations have definite hermiticity assignments and lead to star
structures on the corresponding Hopf algebras. It turns out that for FSM's and
the quantum mechanical time-evolution of QSC's the corresponding stars are
compatible with the Hopf structures. However, in the case of statistical models
the resulting star structure is not a Hopf one but what we call a twisted star.
Real representations of a twisted star Hopf algebra do not close under the
usual tensor product of representations. We briefly comment on the relation of
these results with the Wick rotation. | math-ph |
Numerical Study of the semiclassical limit of the Davey-Stewartson II
equations: We present the first detailed numerical study of the semiclassical limit of
the Davey-Stewartson II equations both for the focusing and the defocusing
variant. We concentrate on rapidly decreasing initial data with a single hump.
The formal limit of these equations for vanishing semiclassical parameter
$\epsilon$, the semiclassical equations, are numerically integrated up to the
formation of a shock. The use of parallelized algorithms allows to determine
the critical time $t_{c}$ and the critical solution for these $2+1$-dimensional
shocks. It is shown that the solutions generically break in isolated points
similarly to the case of the $1+1$-dimensional cubic nonlinear Schr\"odinger
equation, i.e., cubic singularities in the defocusing case and square root
singularities in the focusing case. For small values of $\epsilon$, the full
Davey-Stewartson II equations are integrated for the same initial data up to
the critical time $t_{c}$. The scaling in $\epsilon$ of the difference between
these solutions is found to be the same as in the $1+1$ dimensional case,
proportional to $\epsilon^{2/7}$ for the defocusing case and proportional to
$\epsilon^{2/5}$ in the focusing case. We document the Davey-Stewartson II
solutions for small $\epsilon$ for times much larger than the critical time
$t_{c}$. It is shown that zones of rapid modulated oscillations are formed near
the shocks of the solutions to the semiclassical equations. For smaller
$\epsilon$, the oscillatory zones become smaller and more sharply delimited to
lens shaped regions. Rapid oscillations are also found in the focusing case for
initial data where the singularities of the solution to the semiclassical
equations do not coincide. | math-ph |
Asymptotic Equation for Zeros of Hermite Polynomials from the
Holstein-Primakoff Representation: The Holstein-Primakoff representation for spin systems is used to derive
expressions with solutions that are conjectured to be the zeros of Hermite
polynomials $H_n(x)$ as $n \rightarrow \infty$. This establishes a
correspondence between the zeros of the Hermite polynomials and the boundaries
of the position basis of finite-dimensional Hilbert spaces. | math-ph |
Topological Bifurcations and Reconstruction of Travelling Waves: This paper is devoted to periodic travelling waves solving Lie-Poisson
equations based on the Virasoro group. We show that the reconstruction of any
such solution can be carried out exactly, regardless of the underlying
Hamiltonian (which need not be quadratic), provided the wave belongs to the
coadjoint orbit of a uniform profile. Equivalently, the corresponding "fluid
particle motion" is integrable. Applying this result to the Camassa-Holm
equation, we express the drift of particles in terms of parameters labelling
periodic peakons and exhibit orbital bifurcations: points in parameter space
where the drift velocity varies discontinuously, reflecting a sudden change in
the topology of Virasoro orbits. | math-ph |
Voros Coefficients and the Topological Recursion for a Class of the
Hypergeometric Differential Equations associated with the Degeneration of the
2-dimensional Garnier System: In my joint papers with Iwaki and Koike ([IKoT1, IKoT2]) we found an
intriguing relation between the Voros coefficients in the exact WKB analysis
and the free energy in the topological recursion introduced by Eynard and
Orantin in the case of the confluent family of the Gauss hypergeometric
differential equations. In this paper we discuss its generalization to the case
of the hypergeometric differential equations associated with $2$-dimensional
degenerate Garnier systems. | math-ph |
A triviality result in the AdS/CFT correspondence for Euclidean quantum
fields with exponential interaction: We consider scalar quantum fields with exponential interaction on Euclidean
hyperbolic space $\mathbb{H}^2$ in two dimensions. Using decoupling
inequalities for Neumann boundary conditions on a tessellation of
$\mathbb{H}^2$, we are able to show that the infra-red limit for the generating
functional of the conformal boundary field becomes trivial. | math-ph |
Geometry of Integrable Billiards and Pencils of Quadrics: We study the deep interplay between geometry of quadrics in d-dimensional
space and the dynamics of related integrable billiard systems. Various
generalizations of Poncelet theorem are reviewed. The corresponding analytic
conditions of Cayley's type are derived giving the full description of
periodical billiard trajectories; among other cases, we consider billiards in
arbitrary dimension d with the boundary consisting of arbitrary number k of
confocal quadrics. Several important examples are presented in full details
demonstrating the effectiveness of the obtained results. We give a thorough
analysis of classical ideas and results of Darboux and methodology of Lebesgue,
and prove their natural generalizations, obtaining new interesting properties
of pencils of quadrics. At the same time, we show essential connections between
these classical ideas and the modern algebro-geometric approach in the
integrable systems theory. | math-ph |
Liouville quantum gravity on the annulus: In this work we construct Liouville quantum gravity on an annulus in the
complex plane. This construction is aimed at providing a rigorous mathematical
framework to the work of theoretical physicists initiated by Polyakov in 1981.
It is also a very important example of a conformal field theory (CFT). Results
have already been obtained on the Riemann sphere and on the unit disk so this
paper will follow the same approach. The case of the annulus contains two
difficulties: it is a surface with two boundaries and it has a non-trivial
moduli space. We recover the Weyl anomaly - a formula verified by all CFT - and
deduce from it the KPZ formula. We also show that the full partition function
of Liouville quantum gravity integrated over the moduli space is finite. This
allows us to give the joint law of the Liouville measures and of the random
modulus and to write the conjectured link with random planar maps. | math-ph |
Integrability and separation of variables in Calogero-Coulomb-Stark and
two-center Calogero-Coulomb systems: We propose the integrable N-dimensional Calogero-Coulomb-Stark and two-center
Calogero-Coulomb systems and construct their constants of motion via the Dunkl
operators. Their Schr\"odinger equations decouple in parabolic and elliptic
coordinates into the set of three differential equations like for the
Coulomb-Stark and two-center Coulomb problems. The Calogero term preserves the
energy levels, but changes their degrees of degeneracy. | math-ph |
Modular and Landen transformation between two kinds of separable
solutions of Sine Gordon equation in N=2: In this article, we study on the separable N=2 solutions of Sine Gordon
equation. From the original symmetry,we get two kinds of N=2 separable
solutions. we find these two kinds are related to Landen transformation | math-ph |
Asymptotic Gap Probability Distributions of the Gaussian Unitary
Ensembles and Jacobi Unitary Ensembles: In this paper, we address a class of problems in unitary ensembles.
Specifically, we study the probability that a gap symmetric about 0, i.e.
$(-a,a)$ is found in the Gaussian unitary ensembles (GUE) and the Jacobi
unitary ensembles (JUE) (where in the JUE, we take the parameters
$\alpha=\beta$). By exploiting the even parity of the weight, a doubling of the
interval to $(a^2,\infty)$ for the GUE, and $(a^2,1)$, for the (symmetric) JUE,
shows that the gap probabilities maybe determined as the product of the
smallest eigenvalue distributions of the LUE with parameter $\alpha=-1/2,$ and
$\alpha=1/2$ and the (shifted) JUE with weights $x^{1/2}(1-x)^{\beta}$ and
$x^{-1/2}(1-x)^{\beta}$ The $\sigma$ function, namely, the derivative of the
log of the smallest eigenvalue distributions of the finite-$n$ LUE or the JUE,
satisfies the Jimbo-Miwa-Okamoto $\sigma$ form of $P_{V}$ and $P_{VI}$,
although in the shift Jacobi case, with the weight $x^{\alpha}(1-x)^{\beta},$
the $\beta$ parameter does not show up in the equation. We also obtain the
asymptotic expansions for the smallest eigenvalue distributions of the Laguerre
unitary and Jacobi unitary ensembles after appropriate double scalings, and
obtained the constants in the asymptotic expansion of the gap probablities,
expressed in term of the Barnes $G-$ function valuated at special point. | math-ph |
The relativistic Hopfield model with correlated patterns: In this work we introduce and investigate the properties of the
"relativistic" Hopfield model endowed with temporally correlated patterns.
First, we review the "relativistic" Hopfield model and we briefly describe the
experimental evidence underlying correlation among patterns. Then, we face the
study of the resulting model exploiting statistical-mechanics tools in a
low-load regime. More precisely, we prove the existence of the thermodynamic
limit of the related free-energy and we derive the self-consistence equations
for its order parameters. These equations are solved numerically to get a phase
diagram describing the performance of the system as an associative memory as a
function of its intrinsic parameters (i.e., the degree of noise and of
correlation among patterns). We find that, beyond the standard retrieval and
ergodic phases, the relativistic system exhibits correlated and symmetric
regions -- that are genuine effects of temporal correlation -- whose width is,
respectively, reduced and increased with respect to the classical case. | math-ph |
Superconformal Algebras and Mock Theta Functions 2. Rademacher Expansion
for K3 Surface: The elliptic genera of the K3 surfaces, both compact and non-compact cases,
are studied by using the theory of mock theta functions. We decompose the
elliptic genus in terms of the N=4 superconformal characters at level-1, and
present an exact formula for the coefficients of the massive (non-BPS)
representations using the Poincare-Maass series. | math-ph |
Some integrable systems of algebraic origin and separation of variables: A plane algebraic curve whose Newton polygone contains d lattice points can
be given by d points it passes through. Then the coefficients of its equation
Poisson commute having been regarded as functions of coordinates of those
points. It is observed in the work by O.Babelon and M.Talon, 2002. We formulate
a generalization of this fact in terms of separation of variables and prove
relations implying the Poisson commutativity. The examples of the integrable
systems obtained this way include coefficients of the Lagrange and Hermit
interpolation polynomials, coefficients of the Weierstrass models of curves. | math-ph |
Kosterlitz-Thouless Transition Line for the Two Dimensional Coulomb Gas: With a rigorous renormalization group approach, we study the pressure of the
two dimensional Coulomb Gas along a small piece of the Kosterlitz-Thouless
transition line, i.e. the boundary of the dipole region in the
activity-temperature phase-space. | math-ph |
The orthosymplectic supergroup in harmonic analysis: The orthosymplectic supergroup OSp(m|2n) is introduced as the supergroup of
isometries of flat Riemannian superspace R^{m|2n} which stabilize the origin.
It also corresponds to the supergroup of isometries of the supersphere
S^{m-1|2n}. The Laplace operator and norm squared on R^{m|2n}, which generate
sl(2), are orthosymplectically invariant, therefore we obtain the Howe dual
pair (osp(m|2n),sl(2)). This Howe dual pair solves the problems of the dual
pair (SO(m)xSp(2n),sl(2)), considered in previous papers. In particular we
characterize the invariant functions on flat Riemannian superspace and show
that the integration over the supersphere is uniquely defined by its
orthosymplectic invariance. The supersphere manifold is also introduced in a
mathematically rigorous way. Finally we study the representations of osp(m|2n)
on spherical harmonics. This corresponds to the decomposition of the
supersymmetric tensor space of the m|2n-dimensional super vectorspace under the
action of sl(2)xosp(m|2n). As a side result we obtain information about the
irreducible osp(m|2n)-representations L_{(k,0,...,0)}^{m|2n}. In particular we
find branching rules with respect to osp(m-1|2n). | math-ph |
Exceptional Laguerre and Jacobi polynomials and the corresponding
potentials through Darboux-Crum Transformations: Simple derivation is presented of the four families of infinitely many shape
invariant Hamiltonians corresponding to the exceptional Laguerre and Jacobi
polynomials. Darboux-Crum transformations are applied to connect the well-known
shape invariant Hamiltonians of the radial oscillator and the
Darboux-P\"oschl-Teller potential to the shape invariant potentials of
Odake-Sasaki. Dutta and Roy derived the two lowest members of the exceptional
Laguerre polynomials by this method. The method is expanded to its full
generality and many other ramifications, including the aspects of generalised
Bochner problem and the bispectral property of the exceptional orthogonal
polynomials, are discussed. | math-ph |
Linking numbers in local quantum field theory: Linking numbers appear in local quantum field theory in the presence of
tensor fields, which are closed two-forms on Minkowski space. Given any pair of
such fields, it is shown that the commutator of the corresponding intrinsic
(gauge invariant) vector potentials, integrated about spacelike separated,
spatial loops, are elements of the center of the algebra of all local fields.
Moreover, these commutators are proportional to the linking numbers of the
underlying loops. If the commutators are different from zero, the underlying
two-forms are not exact (there do not exist local vector potentials for them).
The theory then necessarily contains massless particles. A prominent example of
this kind, due to J.E. Roberts, is given by the free electromagnetic field and
its Hodge dual. Further examples with more complex mass spectrum are presented
in this article. | math-ph |
The problem of positivity in 1+1 dimensions and Krein spaces: The possibility of introducing a positive metric on the states of the
massless scalar field in 1+1 dimensions by mean of Krein spaces is examined.
Two different realisations in Krein spaces for the massless scalar field are
compared. It is proved that one is a particular case of the other. The
peculiarities and advantages of both realisations are discussed. | math-ph |
Torsion in Tiling Homology and Cohomology: The first author's recent unexpected discovery of torsion in the integral
cohomology of the T\"ubingen Triangle Tiling has led to a re-evaluation of
current descriptions of and calculational methods for the topological
invariants associated with aperiodic tilings. The existence of torsion calls
into question the previously assumed equivalence of cohomological and
K-theoretic invariants as well as the supposed lack of torsion in the latter.
In this paper we examine in detail the topological invariants of canonical
projection tilings; we extend results of Forrest, Hunton and Kellendonk to give
a full treatment of the torsion in the cohomology of such tilings in
codimension at most 3, and present the additions and amendments needed to
previous results and calculations in the literature. It is straightforward to
give a complete treatment of the torsion components for tilings of codimension
1 and 2, but the case of codimension 3 is a good deal more complicated, and we
illustrate our methods with the calculations of all four icosahedral tilings
previously considered. Turning to the K-theoretic invariants, we show that
cohomology and K-theory agree for all canonical projection tilings in
(physical) dimension at most 3, thus proving the existence of torsion in, for
example, the K-theory of the T\"ubingen Triangle Tiling. The question of the
equivalence of cohomology and K-theory for tilings of higher dimensional
euclidean space remains open. | math-ph |
The Tsallis-Laplace Transform: We introduce here the q-Laplace transform as a new weapon in Tsallis'
arsenal, discussing its main properties and analyzing some examples. The
q-Gaussian instance receives special consideration. Also, we derive the
q-partition function from the q-Laplace transform. | math-ph |
Low-energy spectrum and dynamics of the weakly interacting Bose gas: We consider a gas of N bosons with interactions in the mean-field scaling
regime. We review the proof of an asymptotic expansion of its low-energy
spectrum, eigenstates and dynamics, which provides corrections to Bogoliubov
theory to all orders in 1/N. This is based on joint works with S. Petrat, P.
Pickl, R. Seiringer and A. Soffer. In addition, we derive a full asymptotic
expansion of the ground state one-body reduced density matrix. | math-ph |
Determination of approximate nonlinear self-adjointenss and approximate
conservation law: Approximate nonlinear self-adjointness is an effective method to construct
approximate conservation law of perturbed partial differential equations
(PDEs). In this paper, we study the relations between approximate nonlinear
self-adjointness of perturbed PDEs and nonlinear self-adjointness of the
corresponding unperturbed PDEs, and consequently provide a simple approach to
discriminate approximate nonlinear self-adjointness of perturbed PDEs.
Moreover, a succinct approximate conservation law formula by virtue of the
known conservation law of the unperturbed PDEs is given in an explicit form. As
an application, we classify a class of perturbed wave equations to be
approximate nonlinear self-adjointness and construct the general approximate
conservation laws formulae. The specific examples demonstrate that approximate
nonlinear self-adjointness can generate new approximate conservation laws. | math-ph |
An introduction to spin systems for mathematicians: We give a leisurely, albeit woefully incomplete, overview of quantum field
theory, its relevance to condensed matter systems, and spin systems, which
proceeds via a series of illustrative examples. The goal is to provide readers
from the mathematics community a swift route into recent condensed matter
literature that makes use of topological quantum field theory and ideas from
stable homotopy theory to attack the problem of classification of topological
(or SPT) phases of matter. The toric code and Heisenberg spin chain are briefly
discussed; important conceptual ideas in physics, that may have somehow evaded
discussion for those with purely mathematical training, are also reviewed.
Emphasis is placed on the connection between (algebras of) nonlocal operators
and the appearance of nontrivial TQFTs in the infrared. | math-ph |
Revisiting (quasi-)exactly solvable rational extensions of the Morse
potential: The construction of rationally-extended Morse potentials is analyzed in the
framework of first-order supersymmetric quantum mechanics. The known family of
extended potentials $V_{A,B,{\rm ext}}(x)$, obtained from a conventional Morse
potential $V_{A-1,B}(x)$ by the addition of a bound state below the spectrum of
the latter, is re-obtained. More importantly, the existence of another family
of extended potentials, strictly isospectral to $V_{A+1,B}(x)$, is pointed out
for a well-chosen range of parameter values. Although not shape invariant, such
extended potentials exhibit a kind of `enlarged' shape invariance property, in
the sense that their partner, obtained by translating both the parameter $A$
and the degree $m$ of the polynomial arising in the denominator, belongs to the
same family of extended potentials. The point canonical transformation
connecting the radial oscillator to the Morse potential is also applied to
exactly solvable rationally-extended radial oscillator potentials to build
quasi-exactly solvable rationally-extended Morse ones. | math-ph |
Spreading lengths of Hermite polynomials: The Renyi, Shannon and Fisher spreading lengths of the classical or
hypergeometric orthogonal polynomials, which are quantifiers of their
distribution all over the orthogonality interval, are defined and investigated.
These information-theoretic measures of the associated Rakhmanov probability
density, which are direct measures of the polynomial spreading in the sense of
having the same units as the variable, share interesting properties: invariance
under translations and reflections, linear scaling and vanishing in the limit
that the variable tends towards a given definite value. The expressions of the
Renyi and Fisher lengths for the Hermite polynomials are computed in terms of
the polynomial degree. The combinatorial multivariable Bell polynomials, which
are shown to characterize the finite power of an arbitrary polynomial, play a
relevant role for the computation of these information-theoretic lengths.
Indeed these polynomials allow us to design an error-free computing approach
for the entropic moments (weighted L^q-norms) of Hermite polynomials and
subsequently for the Renyi and Tsallis entropies, as well as for the Renyi
spreading lengths. Sharp bounds for the Shannon length of these polynomials are
also given by means of an information-theoretic-based optimization procedure.
Moreover, it is computationally proved the existence of a linear correlation
between the Shannon length (as well as the second-order Renyi length) and the
standard deviation. Finally, the application to the most popular
quantum-mechanical prototype system, the harmonic oscillator, is discussed and
some relevant asymptotical open issues related to the entropic moments
mentioned previously are posed. | math-ph |
Navier--Stokes equations, the algebraic aspect: Analysis of the Navier-Stokes equations in the frames of the algebraic
approach to systems of partial differential equations (formal theory of
differential equations) is presented. | math-ph |
On the relevance of the differential expressions $f^2+f'^2$, $f+f"$ and
$f f"- f'^2$ for the geometrical and mechanical properties of curves: We present a unified approach to known and new properties of curves by
showing the ubiquity of the expressions in the title in the analytic treatment
of their mechanical and geometric properties | math-ph |
The Lipkin-Meshkov-Glick model as a particular limit of the SU(1,1)
Richardson-Gaudin integrable models: The Lipkin-Meshkov-Glick (LMG) model has a Schwinger boson realization in
terms of a two-level boson pairing Hamiltonian. Through this realization, it
has been shown that the LMG model is a particular case of the SU (1, 1)
Richardson-Gaudin (RG) integrable models. We exploit the exact solvability of
the model tostudy the behavior of the spectral parameters (pairons) that
completely determine the wave function in the different phases, and across the
phase transitions. Based on the relation between the Richardson equations and
the Lam\'e differential equations we develop a method to obtain numerically the
pairons. The dynamics of pairons in the ground and excited states provides new
insights into the first, second and third order phase transitions, as well as
into the crossings taking place in the LMG spectrum. | math-ph |
On the absence of stationary currents: We review proofs of a theorem of Bloch on the absence of macroscopic
stationary currents in quantum systems. The standard proof shows that the
current in 1D vanishes in the large volume limit under rather general
conditions. In higher dimension, the total current across a cross-section does
not need to vanish in gapless systems but it does vanish in gapped systems. We
focus on the latter claim and give a self-contained proof motivated by a
recently introduced index for many-body charge transport in quantum lattice
systems having a conserved $U(1)$-charge. | math-ph |
ABCD Matrices as Similarity Transformations of Wigner Matrices and
Periodic Systems in Optics: The beam transfer matrix, often called the $ABCD$ matrix, is a two-by-two
matrix with unit determinant, and with three independent parameters. It is
noted that this matrix cannot always be diagonalized. It can however be brought
by rotation to a matrix with equal diagonal elements. This equi-diagonal matrix
can then be squeeze-transformed to a rotation, to a squeeze, or to one of the
two shear matrices. It is noted that these one-parameter matrices constitute
the basic elements of the Wigner's little group for space-time symmetries of
elementary particles. Thus every $ABCD$ matrix can be written as a similarity
transformation of one of the Wigner matrices, while the transformation matrix
is a rotation preceded by a squeeze. This mathematical property enables us to
compute scattering processes in periodic systems. Laser cavities and multilayer
optics are discussed in detail. For both cases, it is shown possible to write
the one-cycle transfer matrix as a similarity transformation of one of the
Wigner matrices. It is thus possible to calculate the $ABCD$ matrix for an
arbitrary number of cycles. | math-ph |
Spectra of Laplacian matrices of weighted graphs: structural genericity
properties: This article deals with the spectra of Laplacians of weighted graphs. In this
context, two objects are of fundamental importance for the dynamics of complex
networks: the second eigenvalue of such a spectrum (called algebraic
connectivity) and its associated eigenvector, the so-called Fiedler vector.
Here we prove that, given a Laplacian matrix, it is possible to perturb the
weights of the existing edges in the underlying graph in order to obtain simple
eigenvalues and a Fiedler vector composed of only non-zero entries. These
structural genericity properties with the constraint of not adding edges in the
underlying graph are stronger than the classical ones, for which arbitrary
structural perturbations are allowed. These results open the opportunity to
understand the impact of structural changes on the dynamics of complex systems. | math-ph |
Algebraic Bethe ansatz for Q-operators of the open XXX Heisenberg chain
with arbitrary spin: In this note we construct Q-operators for the spin s open Heisenberg XXX
chain with diagonal boundaries in the framework of the quantum inverse
scattering method. Following the algebraic Bethe ansatz we diagonalise the
introduced Q-operators using the fundamental commutation relations. By acting
on Bethe off-shell states and explicitly evaluating the trace in the auxiliary
space we compute the eigenvalues of the Q-operators in terms of Bethe roots and
show that the unwanted terms vanish if the Bethe equations are satisfied. | math-ph |
Transparent anisotropy for the relaxed micromorphic model: macroscopic
consistency conditions and long wave length asymptotics: In this paper, we study the anisotropy classes of the fourth order elastic
tensors of the relaxed micromorphic model, also introducing their second order
counterpart by using a Voigt-type vector notation. In strong contrast with the
usual micromorphic theories, in our relaxed micromorphic model only classical
elasticity-tensors with at most 21 independent components are studied together
with rotational coupling tensors with at most 6 independent components. We show
that in the limit case $L_c\rightarrow 0$ (which corresponds to considering
very large specimens of a microstructured metamaterial the meso- and
micro-coefficients of the relaxed model can be put in direct relation with the
macroscopic stiffness of the medium via a fundamental homogenization formula.
We also show that a similar homogenization formula is not possible in the case
of the standard Mindlin-Eringen-format of the anisotropic micromorphic model.
Our results allow us to forecast the successful short term application of the
relaxed micromorphic model to the characterization of anisotropic mechanical
metamaterials. | math-ph |
Hyperbolic and Circular Trigonometry and Application to Special
Relativity: We discuss the most elementary properties of the hyperbolic trigonometry and
show how they can be exploited to get a simple, albeit interesting, geometrical
interpretation of the special relativity. It yields indeed a straightforword
understanding of the Lorentz transformation and of the relativistic kinematics
as well. The geometrical framework adopted in the article is useful to disclose
a wealth of alternative trigonometries not taught in undergraduate and graduate
courses. Their introduction could provide an interesting and useful conceptual
tool for students and teachers. | math-ph |
Nonperturbative calculation of Born-Infeld effects on the Schroedinger
spectrum of the hydrogen atom: We present the first nonperturbative numerical calculations of the
nonrelativistic hydrogen spectrum as predicted by first-quantized
electrodynamics with nonlinear Maxwell-Born-Infeld field equations. We also
show rigorous upper and lower bounds on the ground state.
When judged against empirical data our results significantly restrict the
range of viable values of the new electromagnetic constant which is introduced
by the Born-Infeld theory.
We assess Born's own proposal for the value of his constant. | math-ph |
Differential Structure of the Hyperbolic Clifford Algebra: This paper presents a thoughful review of: (a) the Clifford algebra Cl(H_{V})
of multivecfors which is naturally associated with a hyperbolic space H_{V};
(b) the study of the properties of the duality product of multivectors and
multiforms; (c) the theory of k multivector and l multiform variables
multivector extensors over V and (d) the use of the above mentioned structures
to present a theory of the parallelism structure on an arbitrary smooth
manifold introducing the concepts of covariant derivarives, deformed covariant
derivatives and relative covariant derivatives of multivector, multiform fields
and extensors fields. | math-ph |
On the magnetic shield for a Vlasov-Poisson plasma: We study the screening of a bounded body $\Gamma$ against the effect of a
wind of charged particles, by means of a shield produced by a magnetic field
which becomes infinite on the border of $\Gamma$. The charged wind is modeled
by a Vlasov-Poisson plasma, the bounded body by a torus, and the external
magnetic field is taken close to the border of $\Gamma$. We study two models: a
plasma composed by different species with positive or negative charges, and
finite total mass of each species, and another made of many species of the same
sign, each having infinite mass. We investigate the time evolution of both
systems, showing in particular that the plasma particles cannot reach the body.
Finally we discuss possible extensions to more general initial data. We show
also that when the magnetic lines are straight lines, (that imposes an
unbounded body), the previous results can be improved. | math-ph |
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