Hamilton-Jacobi theory in k-cosymplectic field theories
In this paper we extend the geometric formalism of the Hamilton-Jacobi theory for time dependent Mechanics to the case of classical field theories in the k-cosymplectic framework.
math-ph↗
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Application of the asymptotic Taylor expansion method to bistable potentials
A recent method called Asymptotic Taylor expansion (ATEM) is applied to determine the analytical expression for eigenfunctions and numerical results for eigenvalues of the Schrodinger equation for the bistable potentials. Optimal truncation of the Taylor series gives a best possible analytical expression for eigenfunctions and numerical result for eigenvalues. It is shown that the results are obtained by a simple algorithm constructed for a computer system using symbolic or numerical calculation. It is observed that ATEM produces excellent results appropriate with the existing literature.
quant-ph↗
Classical analog of extended phase space SUSY and its breaking
We derive the classical analog of the extended phase space quantum mechanics of the particle with odd degrees of freedom which gives rise to (N=2)-realization of supersymmetry (SUSY) algebra. By means of an iterative procedure, we find the approximate groundstate solutions to the extended Schrödinger-like equation and use these solutions further to calculate the parameters which measure the breaking of extended SUSY such as the groundstate energy. Consequently, we calculate a more practical measure for the SUSY breaking which is the expectation value of an auxiliary field. We analyze non-perturbative mechanism for extended phase space SUSY breaking in the instanton picture and show that this has resulted from tunneling between the classical vacua of the theory. Particular attention is given to the algebraic properties of shape invariance and spectrum generating algebra.
math-ph↗
From Discrete Time Quantum Walk to Continuous Time Quantum Walk in Limit Distribution
The discrete time quantum walk defined as a quantum-mechanical analogue of the discrete time random walk have recently been attracted from various and interdisciplinary fields. In this review, the weak limit theorem, that is, the asymptotic behavior, of the one-dimensional discrete time quantum walk is analytically shown. From the limit distribution of the discrete time quantum walk, the discrete time quantum walk can be taken as the quantum dynamical simulator of some physical systems.
quant-ph↗
On the chaotic character of the stochastic heat equation, before the onset of intermitttency
We consider a nonlinear stochastic heat equation $\partial_tu=\frac{1}{2}\partial_{xx}u+σ(u)\partial_{xt}W$, where $\partial_{xt}W$ denotes space-time white noise and $σ:\mathbf {R}\to \mathbf {R}$ is Lipschitz continuous. We establish that, at every fixed time $t>0$, the global behavior of the solution depends in a critical manner on the structure of the initial function $u_0$: under suitable conditions on $u_0$ and $σ$, $\sup_{x\in \mathbf {R}}u_t(x)$ is a.s. finite when $u_0$ has compact support, whereas with probability one, $\limsup_{|x|\to\infty}u_t(x)/({\log}|x|)^{1/6}>0$ when $u_0$ is bounded uniformly away from zero. This sensitivity to the initial data of the stochastic heat equation is a way to state that the solution to the stochastic heat equation is chaotic at fixed times, well before the onset of intermittency.
math.PR↗
Spectral statistics of Erdős-Rényi graphs I: Local semicircle law
We consider the ensemble of adjacency matrices of Erdős-Rényi random graphs, that is, graphs on $N$ vertices where every edge is chosen independently and with probability $p\equiv p(N)$. We rescale the matrix so that its bulk eigenvalues are of order one. We prove that, as long as $pN\to\infty$ (with a speed at least logarithmic in $N$), the density of eigenvalues of the Erdős-Rényi ensemble is given by the Wigner semicircle law for spectral windows of length larger than $N^{-1}$ (up to logarithmic corrections). As a consequence, all eigenvectors are proved to be completely delocalized in the sense that the $\ell^{\infty}$-norms of the $\ell^2$-normalized eigenvectors are at most of order $N^{-1/2}$ with a very high probability. The estimates in this paper will be used in the companion paper [Spectral statistics of Erdős-Rényi graphs II: Eigenvalue spacing and the extreme eigenvalues (2011) Preprint] to prove the universality of eigenvalue distributions both in the bulk and at the spectral edges under the further restriction that $pN\gg N^{2/3}$.
math.PR↗
Dynamics of condensation in the symmetric inclusion process
The inclusion process is a stochastic lattice gas, which is a natural bosonic counterpart of the well-studied exclusion process and has strong connections to models of heat conduction and applications in population genetics. Like the zero-range process, due to attractive interaction between the particles, the inclusion process can exhibit a condensation transition. In this paper we present first rigorous results on the dynamics of the condensate formation for this class of models. We study the symmetric inclusion process on a finite set $S$ with total number of particles $N$ in the regime of strong interaction, i.e. with independent diffusion rate $m=m_N \to 0$. For the case $N m_N\to\infty$ we show that on the time scale $1/m_N$ condensates emerge from general homogeneous initial conditions, and we precisely characterize their limiting dynamics. In the simplest case of two sites or a fully connected underlying random walk kernel, there is a single condensate hopping over $S$ as a continuous-time random walk. In the non fully connected case several condensates can coexist and exchange mass via intermediate sites in an interesting coarsening process, which consists of a mixture of a diffusive motion and a jump process, until a single condensate is formed. Our result is based on a general two-scale form of the generator, with a fast-scale neutral Wright-Fisher diffusion and a slow-scale deterministic motion. The motion of the condensates is described in terms of the generator of the deterministic motion and the harmonic projection corresponding to the absorbing states of the Wright-Fisher diffusion.
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Notes on entropic characteristics of quantum channels
One of most important issues in quantum information theory concerns transmission of information through noisy quantum channels. We discuss few channel characteristics expressed by means of generalized entropies. Such characteristics can often be dealt in line with more usual treatment based on the von Neumann entropies. For any channel, we show that the $q$-average output entropy of degree $q\geq1$ is bounded from above by the $q$-entropy of the input density matrix. Concavity properties of the $(q,s)$-entropy exchange are considered. Fano type quantum bounds on the $(q,s)$-entropy exchange are derived. We also give upper bounds on the map $(q,s)$-entropies in terms of the output entropy, corresponding to the completely mixed input.
quant-ph↗
Umbral Vade Mecum
In recent years the umbral calculus has emerged from the shadows to provide an elegant correspondence framework that automatically gives systematic solutions of ubiquitous difference equations --- discretized versions of the differential cornerstones appearing in most areas of physics and engineering --- as maps of well-known continuous functions. This correspondence deftly sidesteps the use of more traditional methods to solve these difference equations. The umbral framework is discussed and illustrated here, with special attention given to umbral counterparts of the Airy, Kummer, and Whittaker equations, and to umbral maps of solitons for the Sine-Gordon, Korteweg--de Vries, and Toda systems.
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Information Geometry of Complex Hamiltonians and Exceptional Points
Information geometry provides a tool to systematically investigate parameter sensitivity of the state of a system. If a physical system is described by a linear combination of eigenstates of a complex (that is, non-Hermitian) Hamiltonian, then there can be phase transitions where dynamical properties of the system change abruptly. In the vicinities of the transition points, the state of the system becomes highly sensitive to the changes of the parameters in the Hamiltonian. The parameter sensitivity can then be measured in terms of the Fisher-Rao metric and the associated curvature of the parameter-space manifold. A general scheme for the geometric study of parameter-space manifolds of eigenstates of complex Hamiltonians is outlined here, leading to generic expressions for the metric.
quant-ph↗
A new use of nonlocal symmetries for computing Liouvillian first integrals of rational second order ordinary differential equations
Here we present an efficient method for finding and using a nonlocal symmetry admitted by a rational second order ordinary differential equation (rational 2ODE) in order to find a Liouvillian first integral (belonging to a vast class of Liouvillian functions). In a first stage, we construct an algorithm (improving the methodde veloped in [1]) that computes a nonlocal symmetry of a rational 2ODE. In ase cond stage, based on the knowledge of this symmetry, it is possible to construct three polynomial vector fields (in R2), which "share" the Liouvillian first integral with the rational 2ODE. These "plane" polynomial vector fields can be used to construct a procedure (based on an idea developed in [2]) to determine an integrating factor for the rational 2ODE with a fast probabilistic algorithm. The main advantages of the proposed method are: the obtaining of the nonlocal symmetry is algorithmic and very efficient and, furthermore, its use to find an integrating factor is a sequence of linear or quasilinear processes.
nlin.CD↗
Two-Dimensional Toda-Heisenberg Lattice
We consider a nonlinear model that is a combination of the anisotropic two-dimensional classical Heisenberg and Toda-like lattices. In the framework of the Hirota direct approach, we present the field equations of this model as a bilinear system, which is closely related to the Ablowitz-Ladik hierarchy, and derive its N-soliton solutions.
nlin.SI↗
Euler Equations Related to the Generalized Neveu-Schwarz Algebra
In this paper, we study supersymmetric or bi-superhamiltonian Euler equations related to the generalized Neveu-Schwarz algebra. As an application, we obtain several supersymmetric or bi-superhamiltonian generalizations of some well-known integrable systems including the coupled KdV equation, the 2-component Camassa-Holm equation and the 2-component Hunter-Saxton equation. To our knowledge, most of them are new.
nlin.SI↗
On Addition Formulae for Sigma Functions of Telescopic Curves
A telescopic curve is a certain algebraic curve defined by $m-1$ equations in the affine space of dimension $m$, which can be a hyperelliptic curve and an $(n,s)$ curve as a special case. We extend the addition formulae for sigma functions of $(n,s)$ curves to those of telescopic curves. The expression of the prime form in terms of the derivative of the sigma function is also given.
math.AG↗
A Common Structure in PBW Bases of the Nilpotent Subalgebra of $U_q(\mathfrak{g})$ and Quantized Algebra of Functions
For a finite-dimensional simple Lie algebra $\mathfrak{g}$, let $U^+_q(\mathfrak{g})$ be the positive part of the quantized universal enveloping algebra, and $A_q(\mathfrak{g})$ be the quantized algebra of functions. We show that the transition matrix of the PBW bases of $U^+_q(\mathfrak{g})$ coincides with the intertwiner between the irreducible $A_q(\mathfrak{g})$-modules labeled by two different reduced expressions of the longest element of the Weyl group of $\mathfrak{g}$. This generalizes the earlier result by Sergeev on $A_2$ related to the tetrahedron equation and endows a new representation theoretical interpretation with the recent solution to the 3D reflection equation for $C_2$. Our proof is based on a realization of $U^+_q(\mathfrak{g})$ in a quotient ring of $A_q(\mathfrak{g})$.
math.QA↗
Parameterizing the Simplest Grassmann-Gaussian Relations for Pachner Move 3-3
We consider relations in Grassmann algebra corresponding to the four-dimensional Pachner move 3-3, assuming that there is just one Grassmann variable on each 3-face, and a 4-simplex weight is a Grassmann-Gaussian exponent depending on these variables on its five 3-faces. We show that there exists a large family of such relations; the problem is in finding their algebraic-topologically meaningful parameterization. We solve this problem in part, providing two nicely parameterized subfamilies of such relations. For the second of them, we further investigate the nature of some of its parameters: they turn out to correspond to an exotic analogue of middle homologies. In passing, we also provide the 2-4 Pachner move relation for this second case.
math.QA↗
$c_{\rm eff}$ from Surgery and Modularity
$\widehat{Z}$ invariants, rigorously defined for negative definite plumbed 3-manifolds, are expected - on physical grounds - to exist for every closed, oriented 3-manifold (Gukov et al., 2020). Several prescriptions have been proposed to extend their definition to generic plumbings by reversing the orientation of a negative definite plumbing, thus turning it into a positive definite one (Cheng et al., 2019). Two existing proposals are relevant for this paper: (i) the regularized $+1/r$-surgery conjecture combined with the false-mock modular conjecture (Park, 2021 and Cheng et al., 2024), and (ii) a construction based on resurgence and a false theta function duality (Costin et al., 2023). In this note, we compare these proposals on the class of Brieskorn homology spheres $\overline{Σ(s,t,rst\pm1)}$ and find that they are incompatible in general. Our diagnostic is the effective central charge, $c_{\rm eff}$, which governs the asymptotic growth of coefficients of $\widehat{Z}$ (Gukov and Jagadale, 2024). First, we prove that the upper bound on $c_{\rm eff}$ from prescription (i) is governed by the Ramanujan theta function, which regularizes the surgery formula. Second, we develop numerical and modular tools that deliver the lower bounds as well as exact values via mixed mock-modular analysis. Complementing this, we also study $c_{\rm eff}$ for negative definite plumbed 3-manifolds which allow for a better comparison of pairs of 3-manifolds related by orientation reversal. As a result, we find that for some Brieskorn spheres the surgery and false-mock prescriptions violate the expected relation between $c_{\rm eff}$, Chern-Simons invariants and non-abelian flat connections.These findings underscore $c_{\rm eff}$ as a sensitive probe of the ''positive side'' of $\widehat{Z}$-theory.
hep-th↗
On the hyperbolicity and causality of the relativistic Euler system under the kinetic equation of state
We show that a pair of conjectures raised in [11] concerning the construction of normal solutions to the relativistic Boltzmann equation are valid. This ensures that the results in [11] hold for any range of positive temperatures and that the relativistic Euler system under the kinetic equation of state is hyperbolic and the speed of sound cannot overcome $c/\sqrt{3}$.
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