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Irreducibility of the Fermi Surface for Planar Periodic Graph Operators: We prove that the Fermi surface of a connected doubly periodic self-adjoint
discrete graph operator is irreducible at all but finitely many energies
provided that the graph (1) can be drawn in the plane without crossing edges
(2) has positive coupling coefficients (3) has two vertices per period. If
"positive" is relaxed to "complex", the only cases of reducible Fermi surface
occur for the graph of the tetrakis square tiling, and these can be explicitly
parameterized when the coupling coefficients are real. The irreducibility
result applies to weighted graph Laplacians with positive weights. | math-ph |
Effective su_q(2) models and polynomial algebras for fermion-boson
Hamiltonians: Schematic su(2)+h3 interaction Hamiltonians, where su(2) plays the role of
the pseudo-spin algebra of fermion operators and h3 is the Heisenberg algebra
for bosons, are shown to be closely related to certain nonlinear models defined
on a single quantum algebra q-su(2) of quasifermions. In particular, q-su(2)
analogues of the Da Providencia-Schutte and extended Lipkin models are
presented. The connection between q and the physical parameters of the
fermion-boson system is analysed, and the integrability properties of the
interaction Hamiltonians are discussed by using polynomial algebras. | math-ph |
Riemann Hypothesis, Matrix/Gravity Correspondence and FZZT Brane
Partition Functions: We investigate the physical interpretation of the Riemann zeta function as a
FZZT brane partition function associated with a matrix/gravity correspondence.
The Hilbert-Polya operator in this interpretation is the master matrix of the
large N matrix model. Using a related function $\Xi(z)$ we develop an analogy
between this function and the Airy function Ai(z) of the Gaussian matrix model.
The analogy gives an intuitive physical reason why the zeros lie on a critical
line. Using a Fourier transform of the $\Xi(z)$ function we identify a
Kontsevich integrand. Generalizing this integrand to $n \times n$ matrices we
develop a Kontsevich matrix model which describes n FZZT branes. The Kontsevich
model associated with the $\Xi(z)$ function is given by a superposition of
Liouville type matrix models that have been used to describe matrix model
instantons. | math-ph |
The Continuum Potts Model at the Disorder-Order Transition -- a Study by
Cluster Dynamics: We investigate the continuum q-Potts model at its transition point from the
disordered to the ordered regime, with particular emphasis on the coexistence
of disordered and ordered phases in the high-q case. We argue that occurrence
of phase transition can be seen as percolation in the related random cluster
representation, similarly to the lattice Potts model, and investigate the
typical structure of clusters for high q. We also report on numerical
simulations in two dimensions using a continuum version of the Swendsen-Wang
algorithm, compare the results with earlier simulations which used the invaded
cluster algorithm, and discuss implications on the geometry of clusters in the
disordered and ordered phases. | math-ph |
Scalar products and norm of Bethe vectors for integrable models based on
$U_q(\widehat{\mathfrak{gl}}_{n})$: We obtain recursion formulas for the Bethe vectors of models with periodic
boundary conditions solvable by the nested algebraic Bethe ansatz and based on
the quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_{n})$. We also present
a sum formula for their scalar products. This formula describes the scalar
product in terms of a sum over partitions of the Bethe parameters, whose
factors are characterized by two highest coefficients. We provide different
recursions for these highest coefficients. In addition, we show that when the
Bethe vectors are on-shell, their norm takes the form of a Gaudin determinant. | math-ph |
Maximally extended sl(2|2), q-deformed d(2,1;epsilon) and 3D
kappa-Poincaré: We show that the maximal extension sl(2) times psl(2|2) times C3 of the
sl(2|2) superalgebra can be obtained as a contraction limit of the semi-simple
superalgebra d(2,1;epsilon) times sl(2). We reproduce earlier results on the
corresponding q-deformed Hopf algebra and its universal R-matrix by means of
contraction. We make the curious observation that the above algebra is related
to kappa-Poincar\'e symmetry. When dropping the graded part psl(2|2) we find a
novel one-parameter deformation of the 3D kappa-Poincar\'e algebra. Our
construction also provides a concise exact expression for its universal
R-matrix. | math-ph |
The Global Evolution of States of a Continuum Kawasaki Model with
Repulsion: An infinite system of point particles performing random jumps in
$\mathds{R}^d$ with repulsion is studied. The states of the system are
probability measures on the space of particle's configurations. The result of
the paper is the construction of the global in time evolution of states with
the help of the corresponding correlation functions. It is proved that for each
initial sub-Poissonian state $\mu_0$, the constructed evolution $\mu_0 \mapsto
\mu_t$ preserves this property. That is, $\mu_t$ is sub-Poissonian for all
$t>0$. | math-ph |
Paragrassmann Algebras as Quantum Spaces, Part I: Reproducing Kernels: Paragrassmann algebras are given a sesquilinear form for which one subalgebra
becomes a Hilbert space known as the Segal-Bargmann space. This Hilbert space
as well as the ambient space of the paragrassmann algebra itself are shown to
have reproducing kernels. These algebras are not isomorphic to algebras of
functions so some care must be taken in defining what "evaluation at a point"
corresponds to in this context. The reproducing kernel in the Segal-Bargmann
space is shown to have most, though not all, of the standard properties. These
quantum spaces provide non-trivial examples of spaces which have a reproducing
kernel but which are not spaces of functions. | math-ph |
Super-Poincare' algebras, space-times and supergravities (I): A new formulation of theories of supergravity as theories satisfying a
generalized Principle of General Covariance is given. It is a generalization of
the superspace formulation of simple 4D-supergravity of Wess and Zumino and it
is designed to obtain geometric descriptions for the supergravities that
correspond to the super Poincare' algebras of Alekseevsky and Cortes'
classification. | math-ph |
Metric Reduction in Generalized Geometry and Balanced Topological Field
Theories: The recently established metric reduction in generalized geometry is encoded
in 0-dimensional supersymmetric $\sigma$-models. This is an example of balanced
topological field theories. To find the geometric content of such models, the
reduction of Bismut connections is studies in detail. Generalized
K$\ddot{a}$hler reduction is briefly revisited in this formalism and the
generalized K$\ddot{a}$hler geometry on the moduli space of instantons on a
generalized K$\ddot{a}$hler 4-manifold of even type is thus explained formally
in a topological field theoretic way. | math-ph |
The Batalin-Vilkovisky Formalism and the Determinant Line Bundle: Given a smooth family of massless free fermions parametrized by a base
manifold $B$, we show that the (mathematically rigorous) Batalin-Vilkovisky
quantization of the observables of this family gives rise to the determinant
line bundle for the corresponding family of Dirac operators. | math-ph |
On the Mass Concentration for Bose-Einstein Condensates with Attractive
Interactions: We consider two-dimensional Bose-Einstein condensates with attractive
interaction, described by the Gross-Pitaevskii functional. Minimizers of this
functional exist only if the interaction strength $a$ satisfies $a < a^*=
\|Q\|_2^2$, where $Q$ is the unique positive radial solution of $\Delta
u-u+u^3=0$ in $\R^2$. We present a detailed analysis of the behavior of
minimizers as $a$ approaches $a^*$, where all the mass concentrates at a global
minimum of the trapping potential. | math-ph |
Calculating algebraic entropies: an express method: We describe a method for investigating the integrable character of a given
three-point mapping, provided that the mapping has confined singularities. Our
method, dubbed "express", is inspired by a novel approach recently proposed by
R.G. Halburd. While the latter aims at computing the exact degree growth of a
given mapping based on the structure of its singularities, we content ourselves
with obtaining an answer as to whether a given system is integrable or not. We
present several examples illustrating our method as well as its limitations. We
also compare the present method to the full-deautonomisation approach we
recently introduced. | math-ph |
A simple criterion for essential self-adjointness of Weyl
pseudodifferential operators: We prove new criteria for essential self-adjointness of pseudodifferential
operators which do not involve ellipticity type assumptions. For example, we
show that self-adjointness holds in case that the symbol is $C^{2d+3}$ with
derivatives of order two and higher being uniformly bounded. These results also
apply to hermitian operator-valued symbols on infinite-dimensional Hilbert
spaces which are important to applications in physics. Our method relies on a
phase space differential calculus for quadratic forms on $L^2(\mathbb{R}^d)$,
Calder\'on-Vaillancourt type theorems and a recent self-adjointness result for
Toeplitz operators on the Segal-Bargmann space. | math-ph |
Particle relabelling symmetries and Noether's theorem for vertical slice
models: We consider the variational formulation for vertical slice models introduced
in Cotter and Holm (Proc Roy Soc, 2013). These models have a Kelvin circulation
theorem that holds on all materially-transported closed loops, not just those
loops on isosurfaces of potential temperature. Potential vorticity conservation
can be derived directly from this circulation theorem. In this paper, we show
that this property is due to these models having a relabelling symmetry for
every single diffeomorphism of the vertical slice that preserves the density,
not just those diffeomorphisms that preserve the potential temperature. This is
developed using the methodology of Cotter and Holm (Foundations of
Computational Mathematics, 2012). | math-ph |
A complexity approach to the soliton resolution conjecture: The soliton resolution conjecture is one of the most interesting open
problems in the theory of nonlinear dispersive equations. Roughly speaking it
asserts that a solution with generic initial condition converges to a finite
number of solitons plus a radiative term. In this paper we use the complexity
of a finite object, a notion introduced in Algorithmic Information Theory, to
show that the soliton resolution conjecture is equivalent to the analogous of
the second law of thermodynamics for the complexity of a solution of a
dispersive equation. | math-ph |
Finite Size Corrections for Dimers: In this paper we derive the finite size corrections to the energy eigenvalues
of the energy for 2D dimers on a square lattice. These finite size corrections,
as in the case of Critical Dense Polymers, are proportional to the eigenvalues
of the Local Integrals of Motion of Bazhanov Lukyanov and Zamolodchikov for
central charge $c=-2$. This sheds more light on the status of the Dimer model
as a conformal field theory with this value of the certral charge. | math-ph |
The connection problem associated with a Selberg type integral and the
$q$-Racah polynomials: The connection problem associated with a Selberg type integral is solved. The
connection coefficients are given in terms of the $q$-Racah polynomials. As an
application of the explicit expression of the connection coefficients, examples
of the monodromy-invariant Hermitian form of non-diagonal type are presented.
It is noteworthy that such Hermitian forms are intimately related with the
correlation functions of non-diagonal type in $\hat{sl_2}$-confromal field
theory. | math-ph |
Spectral gap in mean-field $\mathcal O(n)$-model: We study the dependence of the spectral gap for the generator of the
Ginzburg-Landau dynamics for all \emph{$\mathcal O(n)$-models} with mean-field
interaction and magnetic field, below and at the critical temperature on the
number $N$ of particles. For our analysis of the Gibbs measure, we use a
one-step renormalization approach and semiclassical methods to study the
eigenvalue-spacing of an auxiliary Schr\"odinger operator. | math-ph |
The problem of missing terms in term by term integration involving
divergent integrals: Term by term integration may lead to divergent integrals, and naive
evaluation of them by means of, say, analytic continuation or by regularization
or by the finite part integral may lead to missing terms. Here, under certain
analyticity condition, the problem of missing terms for the incomplete
Stieltjes transform, $\int_0^a f(x) (\omega+x)^{-1} \mathrm{d}x$, and the
Stieltjes transform itself, $\int_0^{\infty} f(x) (\omega+x)^{-1} \mathrm{d}x$,
is resolved by lifting the integration in the complex plane. It is shown that
the missing terms arise from the singularities of the complex valued function
$f(z) (\omega + z)^{-1}$, with the divergent integrals arising from term by
term integration interpreted as finite part integrals. | math-ph |
Three types of polynomials related to q-oscillator algebra: This work addresses a full characterization of three new q-polynomials
derived from the $q-$oscillator algebra. Related matrix elements and generating
functions are deduced. Further, a connection between Hahn factorial and
q-Gaussian polynomials is established. | math-ph |
A matrix model of a non-Hermitian $β$-ensemble: We introduce the first random matrix model of a complex $\beta$-ensemble. The
matrices are tridiagonal and can be thought of as the non-Hermitian analogue of
the Hermite $\beta$-ensembles discovered by Dumitriu and Edelman (J. Math.
Phys., Vol. 43, 5830 (2002)). The main feature of the model is that the
exponent $\beta$ of the Vandermonde determinant in the joint probability
density function (j.p.d.f.) of the eigenvalues can take any value in
$\mathbb{R}_+$. However, when $\beta=2$, the j.p.d.f. does not reduce to that
of the Ginibre ensemble, but it contains an extra factor expressed as a
multidimensional integral over the space of the eigenvectors. | math-ph |
On the spectral properties of the Bloch-Torrey equation in infinite
periodically perforated domains: We investigate spectral and asymptotic properties of the particular
Schr\"odinger operator (also known as the Bloch-Torrey operator), $-\Delta + i
g x$, in infinite periodically perforated domains of $\mathbb R^d$. We consider
Dirichlet realizations of this operator and formalize a numerical approach
proposed in [J. Phys. A: Math. Theor. 53, 325201 (2020)] for studying such
operators. In particular, we discuss the existence of the spectrum of this
operator and its asymptotic behavior as $g\to \infty$. | math-ph |
Ordinary differential equations associated with the heat equation: This paper is devoted to the one-dimensional heat equation and the non-linear
ordinary differential equations associated to it.
We consider homogeneous polynomial dynamical systems in the n-dimensional
space, n = 0, 1, 2, .... For any such system our construction matches a
non-linear ordinary differential equation.
We describe the algorithm that brings the solution of such an equation to a
solution of the heat equation. The classical fundamental solution of the heat
equation corresponds to the case n=0 in terms of our construction. Solutions of
the heat equation defined by the elliptic theta-function lead to the Chazy-3
equation and correspond to the case n=2.
The group SL(2, C) acts on the space of solutions of the heat equation. We
show this action for each n induces the action of SL(2, C) on the space of
solutions of the corresponding ordinary differential equations. In the case n=2
this leads to the well-known action of this group on the space of solutions of
the Chazy-3 equation. An explicit description of the family of ordinary
differential equations arising in our approach is given. | math-ph |
Variational equations on mixed Riemannian-Lorentzian metrics: A class of elliptic-hyperbolic equations is placed in the context of a
geometric variational theory, in which the change of type is viewed as a change
in the character of an underlying metric. A fundamental example of a metric
which changes in this way is the extended projective disc, which is Riemannian
at ordinary points, Lorentzian at ideal points, and singular on the absolute.
Harmonic fields on such a metric can be interpreted as the hodograph image of
extremal surfaces in Minkowski 3-space. This suggests an approach to
generalized Plateau problems in 3-dimensional space-time via Hodge theory on
the extended projective disc. Analogous variational problems arise on
Riemannian-Lorentzian flow metrics in fiber bundles (twisted nonlinear Hodge
equations), and on certain singular Riemannian-Lorentzian manifolds which occur
in relativity and quantum cosmology. The examples surveyed come with natural
gauge theories and Hodge dualities. This paper is mainly a review, but some
technical extensions are proven. | math-ph |
Propagation Estimates for Two-cluster Scattering Channels of N-body
Schrödinger Operators: In this paper we prove propagation estimates for two-cluster scattering
channels of N-body Schr\"odinger operators. These estimates are based on the
estimate similar to Mourre's commutator estimate and the method of Skibsted. We
also obtain propagation estimates with better indices using projections onto
almost invariant subspaces close to two-cluster scattering channels. As an
application of these estimates we obtain the resolvent estimate for two-cluster
scattering channels and microlocal propagation estimates in three-body problems
without projections. Our method clearly illustrates evolution of the solutions
of the Schr\"odinger equation. | math-ph |
Dynamics of the infinite discrete nonlinear Schrödinger equation: The discrete nonlinear Schr\"odinger equation on \(\Z^d\), \(d \geq 1\) is an
example of a dispersive nonlinear wave system. Being a Hamiltonian system that
conserves also the \(\ell^2(\Z^d)\)-norm, the well-posedness of the
corresponding Cauchy problem follows for square-summable initial data. In this
paper, we prove that the well-posedness continues to hold for much less regular
initial data, namely anything that has at most a certain power law growth far
away from the origin. The growth condition is loose enough to guarantee that,
at least in dimension \(d=1\), initial data sampled from any reasonable
equilibrium distribution of the defocusing DNLS satisfies it almost surely. | math-ph |
The families of orthogonal, unitary and quaternionic unitary
Cayley--Klein algebras and their central extensions: The families of quasi-simple or Cayley--Klein algebras associated to
antihermitian matrices over R, C and H are described in a unified framework.
These three families include simple and non-simple real Lie algebras which can
be obtained by contracting the pseudo-orthogonal algebras so(p,q) of the Cartan
series $B_l$ and $D_l$, the special pseudo-unitary algebras su(p,q) in the
series $A_l$, and the quaternionic pseudo-unitary algebras sq(p,q) in the
series $C_l$. This approach allows to study many properties for all these Lie
algebras simultaneously. In particular their non-trivial central extensions are
completely determined in arbitrary dimension. | math-ph |
Lagrangian time-discretization of the Hunter-Saxton equation: We study Lagrangian time-discretizations of the Hunter-Saxton equation. Using
the Moser-Veselov approach, we obtain such discretizations defined on the
Virasoro group and on the group of orientation-preserving diffeomorphisms of
the circle. We conjecture that one of these discretizations is integrable. | math-ph |
Scattering of Solitons for Coupled Wave-Particle Equations: We establish a long time soliton asymptotics for a nonlinear system of wave
equation coupled to a charged particle. The coupled system has a six
dimensional manifold of soliton solutions. We show that in the large time
approximation, any solution, with an initial state close to the solitary
manifold, is a sum of a soliton and a dispersive wave which is a solution to
the free wave equation. It is assumed that the charge density satisfies Wiener
condition which is a version of Fermi Golden Rule, and that the momenta of the
charge distribution vanish up to the fourth order. The proof is based on a
development of the general strategy introduced by Buslaev and Perelman:
symplectic projection in Hilbert space onto the solitary manifold, modulation
equations for the parameters of the projection, and decay of the transversal
component. | math-ph |
Geometric Hamiltonian matrix on the analogy between geodesic equation
and Schrödinger equation: By formally comparing the geodesic equation with the Schr\"{o}dinger equation
on Riemannian manifold, we come up with the geometric Hamiltonian matrix on
Riemannian manifold based on the geospin matrix, and we discuss its eigenvalue
equation as well. Meanwhile, we get the geometric Hamiltonian function only
related to the scalar curvature. | math-ph |
On the extended multi-component Toda hierarchy: The extended flow equations of the multi-component Toda hierarchy are
constructed. We give the Hirota bilinear equations and tau function of this new
extended multi-component Toda hierarchy(EMTH). Because of logarithmic terms,
some extended vertex operators are constructed in generalized Hirota bilinear
equations which might be useful in topological field theory and Gromov-Witten
theory. Meanwhile the Darboux transformation and bi-Hamiltonian structure of
this hierarchy are given. From the Hamiltonian tau symmetry, we give another
different tau function of this hierarchy with some unknown mysterious
connections with the one defined from the point of wave functions. | math-ph |
Variational reduction of Hamiltonian systems with general constraints: In the Hamiltonian formalism, and in the presence of a symmetry Lie group, a
variational reduction procedure has already been developed for Hamiltonian
systems without constraints. In this paper we present a procedure of the same
kind, but for the entire class of the higher order constrained systems (HOCS),
described in the Hamiltonian formalism. Last systems include the standard and
generalized nonholonomic Hamiltonian systems as particular cases. When
restricted to Hamiltonian systems without constraints, our procedure gives rise
exactly to the so-called Hamilton-Poincar\'e equations, as expected. In order
to illustrate the procedure, we study in detail the case in which both the
configuration space of the system and the involved symmetry define a trivial
principal bundle. | math-ph |
Filtering of Wide Sense Stationary Quantum Stochastic Processes: We introduce a concept of a quantum wide sense stationary process taking
values in a C*-algebra and expected in a sub-algebra. The power spectrum of
such a process is defined, in analogy to classical theory, as a positive
measure on frequency space taking values in the expected algebra. The notion of
linear quantum filters is introduced as some simple examples mentioned. | math-ph |
Bloch Theory and Quantization of Magnetic Systems: Quantizing the motion of particles on a Riemannian manifold in the presence
of a magnetic field poses the problems of existence and uniqueness of
quantizations. Both of them are settled since the early days of geometric
quantization but there is still some structural insight to gain from spectral
theory. Following the work of Asch, Over & Seiler (1994) for the 2-torus we
describe the relation between quantization on the manifold and Bloch theory on
its covering space for more general compact manifolds. | math-ph |
Anomalous Diffusion in One-Dimensional Disordered Systems: A Discrete
Fractional Laplacian Method: This work extends the applications of Anderson-type Hamiltonians to include
transport characterized by anomalous diffusion. Herein, we investigate the
transport properties of a one-dimensional disordered system that employs the
discrete fractional Laplacian, $(-\Delta)^s,\ s\in(0,2),$ in combination with
results from spectral and measure theory. It is a classical mathematical result
that the standard Anderson model exhibits localization of energy states for all
nonzero disorder in one-dimensional systems. Numerical simulations utilizing
our proposed model demonstrate that this localization effect is enhanced for
sub-diffusive realizations of the operator, $s\in (1,2),$ and that the
super-diffusive realizations of the operator, $s\in (0,1),$ can exhibit energy
states with less localized features. These results suggest that the proposed
method can be used to examine anomalous diffusion in physical systems where
strong interactions, structural defects, and correlated effects are present. | math-ph |
Moments of random quantum marginals via Weingarten calculus: The randomized quantum marginal problem asks about the joint distribution of
the partial traces ("marginals") of a uniform random Hermitian operator with
fixed spectrum acting on a space of tensors. We introduce a new approach to
this problem based on studying the mixed moments of the entries of the
marginals. For randomized quantum marginal problems that describe systems of
distinguishable particles, bosons, or fermions, we prove formulae for these
mixed moments, which determine the joint distribution of the marginals
completely. Our main tool is Weingarten calculus, which provides a method for
computing integrals of polynomial functions with respect to Haar measure on the
unitary group. As an application, in the case of two distinguishable particles,
we prove some results on the asymptotic behavior of the marginals as the
dimension of one or both Hilbert spaces goes to infinity. | math-ph |
Quantum Statistical Mechanics via Boundary Conditions. A Groupoid
Approach to Quantum Spin Systems: We use a groupoid model for the spin algebra to introduce boundary conditions
on quantum spin systems via a Poisson point process representation. We can
describe KMS states of quantum systems by means of a set of equations
resembling the standard DLR equations of classical statistical mechanics. We
introduce a notion of quantum specification which recovers the classical DLR
measures in the particular case of classical interactions. Our results are in
the same direction as those obtained recently by Cha, Naaijkens, and
Nachtergaele, differently somehow from the predicted by Fannes and Werner. | math-ph |
Schroedinger Operators With Few Bound States: We show that whole-line Schr\"odinger operators with finitely many bound
states have no embedded singular spectrum. In contradistinction, we show that
embedded singular spectrum is possible even when the bound states approach the
essential spectrum exponentially fast.
We also prove the following result for one- and two-dimensional Schr\"odinger
operators, $H$, with bounded positive ground states: Given a potential $V$, if
both $H\pm V$ are bounded from below by the ground-state energy of $H$, then
$V\equiv 0$. | math-ph |
Newtonian Flow in Converging-Diverging Capillaries: The one-dimensional Navier-Stokes equations are used to derive analytical
expressions for the relation between pressure and volumetric flow rate in
capillaries of five different converging-diverging axisymmetric geometries for
Newtonian fluids. The results are compared to previously-derived expressions
for the same geometries using the lubrication approximation. The results of the
one-dimensional Navier-Stokes are identical to those obtained from the
lubrication approximation within a non-dimensional numerical factor. The
derived flow expressions have also been validated by comparison to numerical
solutions obtained from discretization with numerical integration. Moreover,
they have been certified by testing the convergence of solutions as the
converging-diverging geometries approach the limiting straight geometry. | math-ph |
Skew orthogonal polynomials for the real and quaternion real Ginibre
ensembles and generalizations: There are some distinguished ensembles of non-Hermitian random matrices for
which the joint PDF can be written down explicitly, is unchanged by rotations,
and furthermore which have the property that the eigenvalues form a Pfaffian
point process. For these ensembles, in which the elements of the matrices are
either real, or real quaternion, the kernel of the Pfaffian is completely
determined by certain skew orthogonal polynomials, which permit an expression
in terms of averages over the characteristic polynomial, and the characteristic
polynomial multiplied by the trace. We use Schur polynomial theory, knowledge
of the value of a Schur polynomial averaged against real, and real quaternion
Gaussian matrices, and the Selberg integral to evaluate these averages. | math-ph |
Fourth order superintegrable systems separating in Polar Coordinates. I.
Exotic Potentials: We present all real quantum mechanical potentials in a two-dimensional
Euclidean space that have the following properties: 1. They allow separation of
variables of the Schr\"odinger equation in polar coordinates, 2. They allow an
independent fourth order integral of motion, 3. It turns out that their angular
dependent part $S(\theta)$ does not satisfy any linear differential equation.
In this case it satisfies a nonlinear ODE that has the Painlev\'e property and
its solutions can be expressed in terms of the Painlev\'e transcendent $P_6$.
We also study the corresponding classical analogs of these potentials. The
polynomial algebra of the integrals of motion is constructed in the classical
case. | math-ph |
On the causality and $K$-causality between measures: Drawing from our earlier works on the notion of causality for nonlocal
phenomena, we propose and study the extension of the Sorkin--Woolgar relation
$K^+$ onto the space of Borel probability measures on a given spacetime. We
show that it retains its fundamental properties of transitivity and closedness.
Furthermore, we list and prove several characterizations of this relation,
including the `nonlocal' analogue of the characterization of $K^+$ in terms of
time functions. This generalizes and casts new light on our earlier results
concerning the causal precedence relation $J^+$ between measures. | math-ph |
Orthogonal and symplectic Yangians and Yang-Baxter R-operators: Yang-Baxter R operators symmetric with respect to the orthogonal and
symplectic algebras are considered in an uniform way. Explicit forms for the
spinorial and metaplectic R operators are obtained. L operators, obeying the
RLL relation with the orthogonal or symplectic fundamental R matrix, are
considered in the interesting cases, where their expansion in inverse powers of
the spectral parameter is truncated. Unlike the case of special linear algebra
symmetry the truncation results in additional conditions on the Lie algebra
generators of which the L operators is built and which can be fulfilled in
distinguished representations only. Further, generalised L operators, obeying
the modified RLL relation with the fundamental R matrix replaced by the
spinorial or metaplectic one, are considered in the particular case of linear
dependence on the spectral parameter. It is shown how by fusion with respect to
the spinorial or metaplectic representation these first order spinorial L
operators reproduce the ordinary L operators with second order truncation. | math-ph |
Affine transformation crossed product like algebras and noncommutative
surfaces: Several classes of *-algebras associated to the action of an affine
transformation are considered, and an investigation of the interplay between
the different classes of algebras is initiated. Connections are established
that relate representations of *-algebras, geometry of algebraic surfaces,
dynamics of affine transformations, graphs and algebras coming from a
quantization procedure of Poisson structures. In particular, algebras related
to surfaces being inverse images of fourth order polynomials (in R^3) are
studied in detail, and a close link between representation theory and geometric
properties is established for compact as well as non-compact surfaces. | math-ph |
Parametric representation of a translation-invariant renormalizable
noncommutative model: We construct here the parametric representation of a translation-invariant
renormalizable scalar model on the noncommutative Moyal space of even dimension
$D$. This representation of the Feynman amplitudes is based on some integral
form of the noncommutative propagator. All types of graphs (planar and
non-planar) are analyzed. The r\^ole played by noncommutativity is explicitly
shown. This parametric representation established allows to calculate the power
counting of the model. Furthermore, the space dimension $D$ is just a parameter
in the formulas obtained. This paves the road for the dimensional
regularization of this noncommutative model. | math-ph |
Bott-Kitaev Periodic Table and the Diagonal Map: Building on the 10-way symmetry classification of disordered fermions, the
authors have recently given a homotopy-theoretic proof of Kitaev's "Periodic
Table" for topological insulators and superconductors. The present paper offers
an introduction to the physical setting and the mathematical model used. Basic
to the proof is the so-called Diagonal Map, a natural transformation akin to
the Bott map of algebraic topology, which increases by one unit both the
momentum-space dimension and the symmetry index of translation-invariant ground
states of gapped free-fermion systems. This mapping is illustrated here with a
few examples of interest. | math-ph |
The theory of contractions of 2D 2nd order quantum superintegrable
systems and its relation to the Askey scheme for hypergeometric orthogonal
polynomials: We describe a contraction theory for 2nd order superintegrable systems,
showing that all such systems in 2 dimensions are limiting cases of a single
system: the generic 3-parameter potential on the 2-sphere, S9 in our listing.
Analogously, all of the quadratic symmetry algebras of these systems can be
obtained by a sequence of contractions starting from S9. By contracting
function space realizations of irreducible representations of the S9 algebra
(which give the structure equations for Racah/Wilson polynomials) to the other
superintegrable systems one obtains the full Askey scheme of orthogonal
hypergeometric polynomials.This relates the scheme directly to explicitly
solvable quantum mechanical systems. Amazingly, all of these contractions of
superintegrable systems with potential are uniquely induced by Wigner Lie
algebra contractions of so(3,C) and e(2,C). The present paper concentrates on
describing this intimate link between Lie algebra and superintegrable system
contractions, with the detailed calculations presented elsewhere. Joint work
with E. Kalnins, S. Post, E. Subag and R. Heinonen | math-ph |
A new kind of geometric phases in open quantum systems and higher gauge
theory: A new approach extending the concept of geometric phases to adiabatic open
quantum systems described by density matrices (mixed states) is proposed. This
new approach is based on an analogy between open quantum systems and
dissipative quantum systems which uses a $C^*$-module structure. The gauge
theory associated with these new geometric phases does not take place in an
usual principal bundle structure but in an higher structure, a categorical
principal bundle (so-called principal 2-bundle or non-abelian bundle gerbes)
which is sometimes a non-abelian twisted bundle. This higher degree in the
gauge theory is a geometrical manifestation of the decoherence induced by the
environment on the quantum system. | math-ph |
Resonant states and classical damping: Using Koopman's approach to classical dynamical systems we show that the
classical damping may be interpreted as appearance of resonant states of the
corresponding Koopman's operator. It turns out that simple classical damped
systems give rise to discrete complex spectra. Therefore, the corresponding
generalized eigenvectors may be interpreted as classical resonant states. | math-ph |
Anyons from Three-Body Hard-Core Interactions in One Dimension: Traditional anyons in two dimensions have generalized exchange statistics
governed by the braid group. By analyzing the topology of configuration space,
we discover that an alternate generalization of the symmetric group governs
particle exchanges when there are hard-core three-body interactions in
one-dimension. We call this new exchange symmetry the traid group and
demonstrate that it has abelian and non-abelian representations that are
neither bosonic nor fermionic, and which also transform differently under
particle exchanges than braid group anyons. We show that generalized exchange
statistics occur because, like hard-core two-body interactions in two
dimensions, hard-core three-body interactions in one dimension create defects
with co-dimension two that make configuration space no longer simply-connected.
Ultracold atoms in effectively one-dimensional optical traps provide a possible
implementation for this alternate manifestation of anyonic physics. | math-ph |
Universality for one-dimensional hierarchical coalescence processes with
double and triple merges: We consider one-dimensional hierarchical coalescence processes (in short
HCPs) where two or three neighboring domains can merge. An HCP consists of an
infinite sequence of stochastic coalescence processes: each process occurs in a
different "epoch" and evolves for an infinite time, while the evolutions in
subsequent epochs are linked in such a way that the initial distribution of
epoch $n+1$ coincides with the final distribution of epoch $n$. Inside each
epoch a domain can incorporate one of its neighboring domains or both of them
if its length belongs to a certain epoch-dependent finite range. Assuming that
the distribution at the beginning of the first epoch is described by a renewal
simple point process, we prove limit theorems for the domain length and for the
position of the leftmost point (if any). Our analysis extends the results
obtained in [Ann. Probab. 40 (2012) 1377-1435] to a larger family of models,
including relevant examples from the physics literature [Europhys. Lett. 27
(1994) 175-180, Phys. Rev. E (3) 68 (2003) 031504]. It reveals the presence of
a common abstract structure behind models which are apparently very different,
thus leading to very similar limit theorems. Finally, we give here a full
characterization of the infinitesimal generator for the dynamics inside each
epoch, thus allowing us to describe the time evolution of the expected value of
regular observables in terms of an ordinary differential equation. | math-ph |
Coideal Quantum Affine Algebra and Boundary Scattering of the Deformed
Hubbard Chain: We consider boundary scattering for a semi-infinite one-dimensional deformed
Hubbard chain with boundary conditions of the same type as for the Y=0 giant
graviton in the AdS/CFT correspondence. We show that the recently constructed
quantum affine algebra of the deformed Hubbard chain has a coideal subalgebra
which is consistent with the reflection (boundary Yang-Baxter) equation. We
derive the corresponding reflection matrix and furthermore show that the
aforementioned algebra in the rational limit specializes to the (generalized)
twisted Yangian of the Y=0 giant graviton. | math-ph |
Coupling of eigenvalues of complex matrices at diabolic and exceptional
points: The paper presents a general theory of coupling of eigenvalues of complex
matrices of arbitrary dimension depending on real parameters. The cases of weak
and strong coupling are distinguished and their geometric interpretation in two
and three-dimensional spaces is given. General asymptotic formulae for
eigenvalue surfaces near diabolic and exceptional points are presented
demonstrating crossing and avoided crossing scenarios. Two physical examples
illustrate effectiveness and accuracy of the presented theory. | math-ph |
Colligative properties of solutions: I. Fixed concentrations: Using the formalism of rigorous statistical mechanics, we study the phenomena
of phase separation and freezing-point depression upon freezing of solutions.
Specifically, we devise an Ising-based model of a solvent-solute system and
show that, in the ensemble with a fixed amount of solute, a macroscopic phase
separation occurs in an interval of values of the chemical potential of the
solvent. The boundaries of the phase separation domain in the phase diagram are
characterized and shown to asymptotically agree with the formulas used in
heuristic analyses of freezing point depression. The limit of infinitesimal
concentrations is described in a subsequent paper. | math-ph |
Shapiro's plane waves in spaces of constant curvature and separation of
variables in real and complex coordinates: The aim of the article to clarify the status of Shapiro plane wave solutions
of the Schr\"odinger's equation in the frames of the well-known general method
of separation of variables. To solve this task, we use the well-known
cylindrical coordinates in Riemann and Lobachevsky spaces, naturally related
with Euler angle-parameters. Conclusion may be drawn: the general method of
separation of variables embraces the all plane wave solutions; the plane waves
in Lobachevsky and Riemann space consist of a small part of the whole set of
basis wave functions of Schr\"odinger equation.
In space of constant positive curvature $S_{3}$, a complex analog of
horospherical coordinates of Lobachevsky space $H_{3}$ is introduced. To
parameterize real space $S_{3}$, two complex coordinates $(r,z)$ must obey
additional restriction in the form of the equation $r^{2} = e^{z-z^{*}} -
e^{2z} $. The metrical tensor of space $S_{3}$ is expressed in terms of $(r,z)$
with additional constraint, or through pairs of conjugate variables $(r,r^{*})$
or $(z,z^{*})$; correspondingly exist three different representations for
Schr\"{o}dinger Hamiltonian. Shapiro plane waves are determined and explored as
solutions of Schr\"odinger equation in complex horosperical coordinates of
$S_{3}$. In particular, two oppositely directed plane waves may be presented as
exponentials in conjugated coordinates. $\Psi_{-}= e^{-\alpha z}$ and
$\Psi_{+}= e^{-\alpha z^{*}}$. Solutions constructed are single-valued, finite,
and continuous functions in spherical space and correspond to discrete energy
levels. | math-ph |
Semiclassical Lp estimates: The purpose of this paper is to use semiclassical analysis to unify and
generalize Lp estimates on high energy eigenfunctions and spectral clusters. In
our approach these estimates do not depend on ellipticity and order, and apply
to operators which are selfadjoint only at the principal level. They are
estimates on weakly approximate solutions to semiclassical pseudodifferential
equations. The revision corrects an exponent in the main theorems. | math-ph |
A Dirac type xp-Model and the Riemann Zeros: We propose a Dirac type modification of the xp-model to a $x \slashed{p}$
model on a semi-infinite cylinder. This model is inspired by recent work by
Sierra et al on the xp-model on the half-line. Our model realizes the
Berry-Keating conjecture on the Riemann zeros. We indicate the connection of
our model to that of gapped graphene with a supercritical Coulomb charge, which
might provide a physical system for the study of the zeros of the Riemann Zeta
function. | math-ph |
Bosonic Laplacians in higher spin Clifford analysis: In this article, we firstly introduce higher spin Clifford analysis, which
are considered as generalizations of classical Clifford analysis by considering
functions taking values in irreducible representations of the spin group. Then,
we introduce a type of second order conformally invariant differential
operators, named as bosonic Laplacians, in the higher spin Clifford analysis.
In particular, we will show their close connections to classical Maxwell
equations. At the end, we will introduce a new perspective to define bosonic
Laplacians, which simplifies the connection between bosonic Laplacians and
Rarita-Schwinger type operators obtained before. Moreover, a matrix type
Rarita-Schwinger operator is obtained and some results related to this new
first order matrix type operator are provided. | math-ph |
Plasma waves reflection from a boundary with specular accommodative
boundary conditions: In the present work the linearized problem of plasma wave reflection from a
boundary of a half--space is solved analytically. Specular accommodative
conditions of plasma wave reflection from plasma boundary are taken into
consideration. Wave reflectance is found as function of the given parameters of
the problem, and its dependence on the normal electron momentum accommodation
coefficient is shown by the authors. The case of resonance when the frequency
of self-consistent electric field oscillations is close to the proper
(Langmuir) plasma oscillations frequency, namely, the case of long wave limit
is analyzed. Refs. 17. Figs. 6. | math-ph |
Fermionic walkers driven out of equilibrium: We consider a discrete-time non-Hamiltonian dynamics of a quantum system
consisting of a finite sample locally coupled to several bi-infinite reservoirs
of fermions with a translation symmetry. In this setup, we compute the
asymptotic state, mean fluxes of fermions into the different reservoirs, as
well as the mean entropy production rate of the dynamics. Formulas are
explicitly expanded to leading order in the strength of the coupling to the
reservoirs. | math-ph |
Entropy and Thermodynamic Temperature in Nonequilibrium Classical
Thermodynamics as Immediate Consequences of the Hahn-Banach Theorem: I.
Existence: The Kelvin-Planck statement of the Second Law of Thermodynamics is a
stricture on the nature of heat receipt by any body suffering a cyclic process.
It makes no mention of temperature or of entropy. Beginning with a
Kelvin-Planck statement of the Second Law, we show that entropy and temperature
-- in particular, existence of functions that relate the local specific entropy
and thermodynamic temperature to the local state in a material body -- emerge
immediately and simultaneously as consequences of the Hahn-Banach Theorem.
Existence of such functions of state requires no stipulation that their domains
be restricted to equilibrium states. Further properties, including uniqueness,
are addressed in a companion paper. | math-ph |
Geometrical theory of diffracted rays, orbiting and complex rays: In this article, the ray tracing method is studied beyond the classical
geometrical theory. The trajectories are here regarded as geodesics in a
Riemannian manifold, whose metric and topological properties are those induced
by the refractive index (or, equivalently, by the potential). First, we derive
the geometrical quantization rule, which is relevant to describe the orbiting
bound-states observed in molecular physics. Next, we derive properties of the
diffracted rays, regarded here as geodesics in a Riemannian manifold with
boundary. A particular attention is devoted to the following problems: (i)
modification of the classical stationary phase method suited to a neighborhood
of a caustic; (ii) derivation of the connection formulae which enable one to
obtain the uniformization of the classical eikonal approximation by patching up
geodesic segments crossing the axial caustic; (iii) extension of the eikonal
equation to mixed hyperbolic-elliptic systems, and generation of complex-valued
rays in the shadow of the caustic. By these methods, we can study the creeping
waves in diffractive scattering, describe the orbiting resonances present in
molecular scattering beside the orbiting bound-states, and, finally, describe
the generation of the evanescent waves, which are relevant in the nuclear
rainbow. | math-ph |
Time dependent delta-prime interactions in dimension one: We solve the Cauchy problem for the Schr\"odinger equation corresponding to
the family of Hamiltonians $H_{\gamma(t)}$ in $L^{2}(\mathbb{R})$ which
describes a $\delta'$-interaction with time-dependent strength $1/\gamma(t)$.
We prove that the strong solution of such a Cauchy problem exits whenever the
map $t\mapsto\gamma(t)$ belongs to the fractional Sobolev space
$H^{3/4}(\mathbb{R})$, thus weakening the hypotheses which would be required by
the known general abstract results. The solution is expressed in terms of the
free evolution and the solution of a Volterra integral equation. | math-ph |
Hypergeometric integrals, hook formulas and Whittaker vectors: We determine the coefficient of proportionality between two multidimensional
hypergeometric integrals. One of them is a solution of the dynamical difference
equations associated with a Young diagram and the other is the vertex integral
associated with the Young diagram. The coefficient of proportionality is the
inverse of the product of weighted hooks of the Young diagram. It turns out
that this problem is closely related to the question of describing the action
of the center of the universal enveloping algebra of $\mathfrak{gl}_n$ on the
space of Whittaker vectors in the tensor product of dual Verma modules with
fundamental modules, for which we give an explicit basis of simultaneous
eigenvectors. | math-ph |
Exact solution of the XXX Gaudin model with the generic open boundaries: The XXX Gaudin model with generic integrable boundaries specified by the most
general non-diagonal K-matrices is studied by the off-diagonal Bethe ansatz
method. The eigenvalues of the associated Gaudin operators and the
corresponding Bethe ansatz equations are obtained. | math-ph |
Structure of Clifford-Weyl Algebras And Representations of
ortho-symplectic Lie Superalgebras: In this article, the structure of the Clifford-Weyl superalgebras and their
associated Lie superalgebras will be investigated. These superalgebras have a
natural supersymmetric inner product which is invariant under their Lie
superalgebra structures. The Clifford-Weyl superalgebras can be realized as
tensor product of the algebra of alternating and symmetric tensors
respectively, on the even and odd parts of their underlying superspace. For
Physical applications in elementary particles, we add star structures to these
algebras and investigate the basic relations. Ortho-symplectic Lie algebras are
naturally present in these algebras and their representations on these algebras
can be described easily. | math-ph |
Mean-field limit of Bose systems: rigorous results: We review recent results about the derivation of the Gross-Pitaevskii
equation and of the Bogoliubov excitation spectrum, starting from many-body
quantum mechanics. We focus on the mean-field regime, where the interaction is
multiplied by a coupling constant of order 1/N where N is the number of
particles in the system. | math-ph |
Pressure Derivative on Uncountable Alphabet Setting: a Ruelle Operator
Approach: In this paper we use a recent version of the Ruelle-Perron-Frobenius Theorem
to compute, in terms of the maximal eigendata of the Ruelle operator, the
pressure derivative of translation invariant spin systems taking values on a
general compact metric space. On this setting the absence of metastable states
for continuous potentials on one-dimensional one-sided lattice is proved. We
apply our results, to show that the pressure of an essentially one-dimensional
Heisenberg-type model, on the lattice $\mathbb{N}\times \mathbb{Z}$, is
Fr\'echet differentiable, on a suitable Banach space. Additionally, exponential
decay of the two-point function, for this model, is obtained for any positive
temperature. | math-ph |
Energetic Variational Approaches for inviscid multiphase flow systems
with surface flow and tension: We consider the governing equations for the motion of the inviscid fluids in
two moving domains and an evolving surface from an energetic point of view. We
employ our energetic variational approaches to derive inviscid multiphase flow
systems with surface flow and tension. More precisely, we calculate the
variation of the flow maps to the action integral for our model to derive both
surface flow and tension. We also study the conservation and energy laws of our
multiphase flow systems. The key idea of deriving the pressure of the
compressible fluid on the surface is to make use of the feature of the
barotropic fluid, and the key idea of deriving the pressure of the
incompressible fluid on the surface is to apply a generalized Helmholtz-Weyl
decomposition on a closed surface. | math-ph |
On the eigenvalue problem for arbitrary odd elements of the Lie
superalgebra gl(1|n) and applications: In a Wigner quantum mechanical model, with a solution in terms of the Lie
superalgebra gl(1|n), one is faced with determining the eigenvalues and
eigenvectors for an arbitrary self-adjoint odd element of gl(1|n) in any
unitary irreducible representation W. We show that the eigenvalue problem can
be solved by the decomposition of W with respect to the branching gl(1|n) -->
gl(1|1) + gl(n-1). The eigenvector problem is much harder, since the
Gel'fand-Zetlin basis of W is involved, and the explicit actions of gl(1|n)
generators on this basis are fairly complicated. Using properties of the
Gel'fand-Zetlin basis, we manage to present a solution for this problem as
well. Our solution is illustrated for two special classes of unitary gl(1|n)
representations: the so-called Fock representations and the ladder
representations. | math-ph |
A Simple Derivation of Josephson Formulae in Superconductivity: A simple and general derivation of Josephson formulae for the tunneling
currents is presented on the basis of Sewell's general formulation of
superconductivity in use of off-diagonal long range order (ODLRO). | math-ph |
Integrable three-state vertex models with weights lying on genus five
curves: We investigate the Yang-Baxter algebra for $\mathrm{U}(1)$ invariant
three-state vertex models whose Boltzmann weights configurations break
explicitly the parity-time reversal symmetry. We uncover two families of
regular Lax operators with nineteen non-null weights which ultimately sit on
algebraic plane curves with genus five. We argue that these curves admit degree
two morphisms onto elliptic curves and thus they are bielliptic. The associated
$\mathrm{R}$-matrices are non-additive in the spectral parameters and it has
been checked that they satisfy the Yang-Baxter equation. The respective
integrable quantum spin-1 Hamiltonians are exhibited. | math-ph |
On the mean-field equations for ferromagnetic spin systems: We derive mean-field equations for a general class of ferromagnetic spin
systems with an explicit error bound in finite volumes. The proof is based on a
link between the mean-field equation and the free convolution formalism of
random matrix theory, which we exploit in terms of a dynamical method. We
present three sample applications of our results to Ka\'{c} interactions,
randomly diluted models, and models with an asymptotically vanishing external
field. | math-ph |
Principal Eigenvalue and Landscape Function of the Anderson Model on a
Large Box: We state a precise formulation of a conjecture concerning the product of the
principal eigenvalue and the sup-norm of the landscape function of the Anderson
model restricted to a large box. We first provide the asymptotic of the
principal eigenvalue as the size of the box grows and then use it to give a
partial proof of the conjecture. We give a complete proof for the one
dimensional case. | math-ph |
General bulk-edge correspondence at positive temperature: By extending the gauge covariant magnetic perturbation theory to operators
defined on half-planes, we prove that for $2d$ random ergodic magnetic
Schr\"odinger operators, the celebrated bulk-edge correspondence can be
obtained from a general bulk-edge duality at positive temperature involving the
bulk magnetization and the total edge current.
Our main result is encapsulated in a formula, which states that the
derivative of a large class of bulk partition functions with respect to the
external constant magnetic field, equals the expectation of a corresponding
edge distribution function of the velocity component which is parallel to the
edge. Neither spectral gaps, nor mobility gaps, nor topological arguments are
required.
The equality between the bulk and edge indices, as stated by the conventional
bulk-edge correspondence, is obtained as a corollary of our purely analytical
arguments by imposing a gap condition and by taking a ``zero-temperature"
limit. | math-ph |
Carleman estimates for global uniqueness, stability and numerical
methods for coefficient inverse problems: This is a review paper of the role of Carleman estimates in the theory of
Multidimensional Coefficient Inverse Problems since the first inception of this
idea in 1981. | math-ph |
On Grover's Search Algorithm from a Quantum Information Geometry
Viewpoint: We present an information geometric characterization of Grover's quantum
search algorithm. First, we quantify the notion of quantum distinguishability
between parametric density operators by means of the Wigner-Yanase quantum
information metric. We then show that the quantum searching problem can be
recast in an information geometric framework where Grover's dynamics is
characterized by a geodesic on the manifold of the parametric density operators
of pure quantum states constructed from the continuous approximation of the
parametric quantum output state in Grover's algorithm. We also discuss possible
deviations from Grover's algorithm within this quantum information geometric
setting. | math-ph |
Interpolating between Rényi entanglement entropies for arbitrary
bipartitions via operator geometric means: The asymptotic restriction problem for tensors can be reduced to finding all
parameters that are normalized, monotone under restrictions, additive under
direct sums and multiplicative under tensor products, the simplest of which are
the flattening ranks. Over the complex numbers, a refinement of this problem,
originating in the theory of quantum entanglement, is to find the optimal rate
of entanglement transformations as a function of the error exponent. This
trade-off can also be characterized in terms of the set of normalized,
additive, multiplicative functionals that are monotone in a suitable sense,
which includes the restriction-monotones as well. For example, the flattening
ranks generalize to the (exponentiated) R\'enyi entanglement entropies of order
$\alpha\in[0,1]$. More complicated parameters of this type are known, which
interpolate between the flattening ranks or R\'enyi entropies for special
bipartitions, with one of the parts being a single tensor factor.
We introduce a new construction of subadditive and submultiplicative
monotones in terms of a regularized R\'enyi divergence between many copies of
the pure state represented by the tensor and a suitable sequence of positive
operators. We give explicit families of operators that correspond to the
flattening-based functionals, and show that they can be combined in a
nontrivial way using weighted operator geometric means. This leads to a new
characterization of the previously known additive and multiplicative monotones,
and gives new submultiplicative and subadditive monotones that interpolate
between the R\'enyi entropies for all bipartitions. We show that for each such
monotone there exist pointwise smaller multiplicative and additive ones as
well. In addition, we find lower bounds on the new functionals that are
superadditive and supermultiplicative. | math-ph |
Polynomial solutions of certain differential equations arising in
physics: Linear differential equations of arbitrary order with polynomial coefficients
are considered. Specifically, necessary and sufficient conditions for the
existence of polynomial solutions of a given degree are obtained for these
equations. An algorithm to determine these conditions and to construct the
polynomial solutions is given. The effectiveness of this algorithmic approach
is illustrated by applying it to several differential equations that arise in
mathematical physics. | math-ph |
Singular solutions to the Seiberg-Witten and Freund equations on flat
space from an iterative method: Although it is well known that the Seiberg-Witten equations do not admit
nontrivial $L^2$ solutions in flat space, singular solutions to them have been
previously exhibited -- either in $R^3$ or in the dimensionally reduced spaces
$R^2$ and $R^1$ -- which have physical interest. In this work, we employ an
extension of the Hopf fibration to obtain an iterative procedure to generate
particular singular solutions to the Seiberg-Witten and Freund equations on
flat space. Examples of solutions obtained by such method are presented and
briefly discussed. | math-ph |
Evaluation on asymptotic distribution of particle systems expressed by
probabilistic cellular automata: We propose some conjectures for asymptotic distribution of probabilistic
Burgers cellular automaton (PBCA) which is defined by a simple motion rule of
particles including a probabilistic parameter. Asymptotic distribution of
configurations converges to a unique steady state for PBCA. We assume some
conjecture on the distribution and derive the asymptotic probability expressed
by GKZ hypergeometric function. If we take a limit of space size to infinity, a
relation between density and flux of particles for infinite space size can be
evaluated. Moreover, we propose two extended systems of PBCA of which
asymptotic behavior can be analyzed as PBCA. | math-ph |
On the spectrum of the Laplace operator of metric graphs attached at a
vertex -- Spectral determinant approach: We consider a metric graph $\mathcal{G}$ made of two graphs $\mathcal{G}_1$
and $\mathcal{G}_2$ attached at one point. We derive a formula relating the
spectral determinant of the Laplace operator
$S_\mathcal{G}(\gamma)=\det(\gamma-\Delta)$ in terms of the spectral
determinants of the two subgraphs. The result is generalized to describe the
attachment of $n$ graphs. The formulae are also valid for the spectral
determinant of the Schr\"odinger operator $\det(\gamma-\Delta+V(x))$. | math-ph |
Heat Current Properties of a Rotor Chain Type Model with
Next-Nearest-Neighbor Interactions: In this article, to study the heat flow behavior, we perform analytical
investigations in a rotor chain type model (involving inner stochastic noises)
with next and next-nearest-neighbor interactions. It is known in the literature
that the chain rotor model with long range interactions presents an insulating
phase for the heat conductivity. But we show, in contrast with such a behavior,
that the addition of a next-nearest-neighbor potential increases the thermal
conductivity, at least in the low temperature regime, indicating that the
insulating property is a genuine long range interaction effect. We still
establish, now by numerical computations, the existence of a thermal
rectification in systems with graded structures. | math-ph |
Counting functions for branched covers of elliptic curves and
quasi-modular forms: We prove that each counting function of the m-simple branched covers with a
fixed genus of an elliptic curve is expressed as a polynomial of the Eisenstein
series E_2, E_4 and E_6 . The special case m=2 is considered by Dijkgraaf. | math-ph |
On a class of second-order PDEs admitting partner symmetries: Recently we have demonstrated how to use partner symmetries for obtaining
noninvariant solutions of heavenly equations of Plebanski that govern heavenly
gravitational metrics. In this paper, we present a class of scalar second-order
PDEs with four variables, that possess partner symmetries and contain only
second derivatives of the unknown. We present a general form of such a PDE
together with recursion relations between partner symmetries. This general PDE
is transformed to several simplest canonical forms containing the two heavenly
equations of Plebanski among them and two other nonlinear equations which we
call mixed heavenly equation and asymmetric heavenly equation. On an example of
the mixed heavenly equation, we show how to use partner symmetries for
obtaining noninvariant solutions of PDEs by a lift from invariant solutions.
Finally, we present Ricci-flat self-dual metrics governed by solutions of the
mixed heavenly equation and its Legendre transform. | math-ph |
On the partition function of the $Sp(2n)$ integrable vertex model: We study the partition function per site of the integrable $Sp(2n)$ vertex
model on the square lattice. We establish a set of transfer matrix fusion
relations for this model. The solution of these functional relations in the
thermodynamic limit allows us to compute the partition function per site of the
fundamental $Sp(2n)$ representation of the vertex model. In addition, we also
obtain the partition function of vertex models mixing the fundamental with
other representations. | math-ph |
Bulk and Boundary Invariants for Complex Topological Insulators: From
K-Theory to Physics: This monograph offers an overview on the topological invariants in fermionic
topological insulators from the complex classes. Tools from K-theory and
non-commutative geometry are used to define bulk and boundary invariants, to
establish the bulk-boundary correspondence and to link the invariants to
physical observables. | math-ph |
Coordinate-free description of corrugated flames with realistic density
drop at the front: The complete set of hydrodynamic equations for a corrugated flame front is
reduced to a system of coordinate-free equations at the front using the fact
that vorticity effects remain relatively weak even for corrugated flames. It is
demonstrated how small but finite flame thickness may be taken into account in
the equations. Similar equations are obtained for turbulent burning in the
flamelet regime. The equations for a turbulent corrugated flame are consistent
with the Taylor hypothesis of stationary external turbulence. | math-ph |
Evaluation of Spectral Zeta-Functions with the Renormalization Group: We evaluate spectral zeta-functions of certain network Laplacians that can be
treated exactly with the renormalization group. As specific examples we
consider a class of Hanoi networks and those hierarchical networks obtained by
the Migdal-Kadanoff bond moving scheme from regular lattices. As possible
applications of these results we mention quantum search algorithms as well as
synchronization, which we discuss in more detail. | math-ph |
Quantum energy inequalities and local covariance II: Categorical
formulation: We formulate Quantum Energy Inequalities (QEIs) in the framework of locally
covariant quantum field theory developed by Brunetti, Fredenhagen and Verch,
which is based on notions taken from category theory. This leads to a new
viewpoint on the QEIs, and also to the identification of a new structural
property of locally covariant quantum field theory, which we call Local
Physical Equivalence. Covariant formulations of the numerical range and
spectrum of locally covariant fields are given and investigated, and a new
algebra of fields is identified, in which fields are treated independently of
their realisation on particular spacetimes and manifestly covariant versions of
the functional calculus may be formulated. | math-ph |
Tosio Kato's Work on Non--Relativistic Quantum Mechanics: We review the work of Tosio Kato on the mathematics of non--relativistic
quantum mechanics and some of the research that was motivated by this. Topics
include analytic and asymptotic eigenvalue perturbation theory, Temple--Kato
inequality, self--adjointness results, quadratic forms including monotone
convergence theorems, absence of embedded eigenvalues, trace class scattering,
Kato smoothness, the quantum adiabatic theorem and Kato's ultimate Trotter
Product Formula. | math-ph |
Virasoro constraints and polynomial recursion for the linear Hodge
integrals: The Hodge tau-function is a generating function for the linear Hodge
integrals. It is also a tau-function of the KP hierarchy. In this paper, we
first present the Virasoro constraints for the Hodge tau-function in the
explicit form of the Virasoro equations. The expression of our Virasoro
constraints is simply a linear combination of the Virasoro operators, where the
coefficients are restored from a power series for the Lambert W function. Then,
using this result, we deduce a simple version of the Virasoro constraints for
the linear Hodge partition function, where the coefficients are restored from
the Gamma function. Finally, we establish the equivalence relation between the
Virasoro constraints and polynomial recursion formula for the linear Hodge
integrals. | math-ph |
Determinant expressions of constraint polynomials and the spectrum of
the asymmetric quantum Rabi model: The purpose of this paper is to study the exceptional eigenvalues of the
asymmetric quantum Rabi models (AQRM), specifically, to determine the
degeneracy of their eigenstates. Here, the Hamiltonian
$H^{\epsilon}_{\text{Rabi}}$ of the AQRM is defined by adding the fluctuation
term $\epsilon \sigma_x$, with $\sigma_x$ being the Pauli matrix, to the
Hamiltonian of the quantum Rabi model, breaking its $\mathbb{Z}_{2}$-symmetry.
The spectrum of $H^{\epsilon}_{\text{Rabi}}$ contains a set of exceptional
eigenvalues, considered to be remains of the eigenvalues of the uncoupled
bosonic mode, which are further classified in two types: Juddian, associated
with polynomial eigensolutions, and non-Juddian exceptional. We explicitly
describe the constraint relations for allowing the model to have exceptional
eigenvalues. By studying these relations we obtain the proof of the conjecture
on constraint polynomials previously proposed by the third author. In fact, we
prove that the spectrum of the AQRM possesses degeneracies if and only if the
parameter $\epsilon$ is a half-integer. Moreover, we show that non-Juddian
exceptional eigenvalues do not contribute any degeneracy and we characterize
exceptional eigenvalues by representations of $\mathfrak{sl}_2$. Upon these
results, we draw the whole picture of the spectrum of the AQRM. Furthermore,
generating functions of constraint polynomials from the viewpoint of confluent
Heun equations are also discussed. | math-ph |
Droplet minimizers for the Gates-Lebowitz-Penrose free energy functional: We study the structure of the constrained minimizers of the
Gates-Lebowitz-Penrose free-energy functional ${\mathcal F}_{\rm GLP}(m)$,
non-local functional of a density field $m(x)$, $x\in {\mathcal T}_L$, a
$d$-dimensional torus of side length $L$. At low temperatures, ${\mathcal
F}_{\rm GLP}$ is not convex, and has two distinct global minimizers,
corresponding to two equilibrium states. Here we constrain the average density
$L^{-d}\int_{{\cal T}_L}m(x)\dd x$ to be a fixed value $n$ between the
densities in the two equilibrium states, but close to the low density
equilibrium value. In this case, a "droplet" of the high density phase may or
may not form in a background of the low density phase, depending on the values
$n$ and $L$. We determine the critical density for droplet formation, and the
nature of the droplet, as a function of $n$ and $L$. The relation between the
free energy and the large deviations functional for a particle model with
long-range Kac potentials, proven in some cases, and expected to be true in
general, then provides information on the structure of typical microscopic
configurations of the Gibbs measure when the range of the Kac potential is
large enough. | math-ph |
Nodal count of graph eigenfunctions via magnetic perturbation: We establish a connection between the stability of an eigenvalue under a
magnetic perturbation and the number of zeros of the corresponding
eigenfunction. Namely, we consider an eigenfunction of discrete Laplacian on a
graph and count the number of edges where the eigenfunction changes sign (has a
"zero"). It is known that the $n$-th eigenfunction has $n-1+s$ such zeros,
where the "nodal surplus" $s$ is an integer between 0 and the number of cycles
on the graph.
We then perturb the Laplacian by a weak magnetic field and view the $n$-th
eigenvalue as a function of the perturbation. It is shown that this function
has a critical point at the zero field and that the Morse index of the critical
point is equal to the nodal surplus $s$ of the $n$-th eigenfunction of the
unperturbed graph. | math-ph |
The conditional DPP approach to random matrix distributions: We present the conditional determinantal point process (DPP) approach to
obtain new (mostly Fredholm determinantal) expressions for various eigenvalue
statistics in random matrix theory. It is well-known that many (especially
$\beta=2$) eigenvalue $n$-point correlation functions are given in terms of
$n\times n$ determinants, i.e., they are continuous DPPs. We exploit a derived
kernel of the conditional DPP which gives the $n$-point correlation function
conditioned on the event of some eigenvalues already existing at fixed
locations.
Using such kernels we obtain new determinantal expressions for the joint
densities of the $k$ largest eigenvalues, probability density functions of the
$k^\text{th}$ largest eigenvalue, density of the first eigenvalue spacing, and
more. Our formulae are highly amenable to numerical computations and we provide
various numerical experiments. Several numerical values that required hours of
computing time could now be computed in seconds with our expressions, which
proves the effectiveness of our approach.
We also demonstrate that our technique can be applied to an efficient
sampling of DR paths of the Aztec diamond domino tiling. Further extending the
conditional DPP sampling technique, we sample Airy processes from the extended
Airy kernel. Additionally we propose a sampling method for non-Hermitian
projection DPPs. | math-ph |
Internal Lagrangians of PDEs as variational principles: A description of how the principle of stationary action reproduces itself in
terms of the intrinsic geometry of variational equations is proposed. A notion
of stationary points of an internal Lagrangian is introduced. A connection
between symmetries, conservation laws and internal Lagrangians is established.
Noether's theorem is formulated in terms of internal Lagrangians. A relation
between non-degenerate Lagrangians and the corresponding internal Lagrangians
is investigated. Several examples are discussed. | math-ph |
Heat Kernels and Zeta Functions on Fractals: On fractals, spectral functions such as heat kernels and zeta functions
exhibit novel features, very different from their behaviour on regular smooth
manifolds, and these can have important physical consequences for both
classical and quantum physics in systems having fractal properties. | math-ph |
Elliptic Calogero-Moser system, crossed and folded instantons, and
bilinear identities: Affine analogues of the Q-functions are constructed using folded instantons
partition functions. They are shown to be the solutions of the quantum spectral
curve of the N-body elliptic Calogero-Moser (eCM) system, the quantum Krichever
curve. They also solve the elliptic analogue of the quantum Wronskian equation.
In the companion paper we present the quantum analogue of Krichever's Lax
operator for eCM. A connection to crossed instantons on Taub-Nut spaces, and
opers on a punctured torus is pointed out. | math-ph |
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