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In this paper, we obtain the long-time asymptotics of the complex mKdV equation via the Defit–Zhou method (nonlinear steepest descent method). The Cauchy problem of the complex mKdV equation is transformed into the corresponding Riemann–Hilbert problem on the basis of the Lax pair and the scattering matrix. After that, Riemann–Hilbert problems are converted through a decomposition of the matrix-valued spectral function and factorizations of the jump matrix for Riemann–Hilbert problem. Finally, by solving the last model problem, the long-time asymptotics of complex mKdV equation are derived. [ABSTRACT FROM AUTHOR]

We consider the expansion problem of a wedge of gas into vacuum in the two-dimensional isentropic Euler equation for a class of pressure laws. We prove the global existence of the smooth solution in the case 0 < θ < 2 θ ̄ ( ρ 0 ) , where θ is the half-angle of the wedge, and simplify this problem to a two-dimensional self-similar Euler system with a Goursat problem. By using a special symmetric structure of the characteristic form as well as characteristic decompositions, a prior C1 estimate of the solution of the Goursat problem can be obtained. [ABSTRACT FROM AUTHOR]

We study solutions to Nahm's equations with continuous symmetries, and under certain (mild) hypotheses, we classify the corresponding Ansätze. Using our classification, we construct novel Nahm data and prescribe methods for generating further solutions. Finally, we use these results to construct new BPS monopoles with spherical symmetry. [ABSTRACT FROM AUTHOR]

We associate a deformation of the Heisenberg algebra with the suitably normalized Yang R-matrix, and we investigate its properties. Moreover, we construct new examples of quantum vertex algebras, which possess the same representation theory as the aforementioned deformed Heisenberg algebra. [ABSTRACT FROM AUTHOR]

Due to the fact that the higher-order Kaup–Newell (KN) system has more complex and diverse solutions than the classical second-order flow KN system, the research on it has attracted much attention. In this paper, we consider a higher-order KN equation with third-order dispersion and fifth-order nonlinearity. Based on the theory of the inverse scattering, the matrix Riemann–Hilbert problem is established. Through the dressing method, the solution matrix with simple zeros without reflection is constructed. In particular, a new form of solution is given which is more direct and simpler than previous methods. In addition, through the determinant solution matrix, the vivid diagrams and dynamic analysis of single-soliton solution and two-soliton solution are given in detail. Finally, by using the technique of limit, we construct the general solution matrix in the case of multiple zeros, and the examples of solutions for the cases of double zeros, triple zeros, single–double zeros, and double–double zeros are especially shown. [ABSTRACT FROM AUTHOR]

Universality vs non-universality on the quantum Zeno effect and the anti-Zeno effect is discussed. It is well-known that the Zeno effect is a result of the branch point effect based on the constraint that the energy spectrum is bounded from below. On the other hand, we show that the anti-Zeno effect arises from the interference of multiple complex poles in the lower half energy plane on a larger time scale compared with the Zeno time. Since the production of extra complex poles depends on the characteristics of the interaction Hamiltonian, we emphasize that the anti-Zeno effect is model-dependent and non-universal, while the Zeno effect is a universal phenomenon independent of the form of the interaction. [ABSTRACT FROM AUTHOR]

The first work of Dyson relating to random matrix theory, "The dynamics of a disordered linear chain," is reviewed. Contained in this work is an exact solution of the so-called type I chain in the case of the disorder variables being given by a gamma distribution. The exact solution exhibits a singularity in the density of states about the origin, which has since been shown to be universal for one-dimensional tight binding models with off diagonal disorder. We discuss this context and also point out some universal features of the weak disorder expansion of the exact solution near the band edge. Furthermore, a link between the exact solution and a tridiagonal formalism of anti-symmetric Gaussian β-ensembles with β proportional to 1/N is made. [ABSTRACT FROM AUTHOR]

In this work, we deal with the dynamics of a ferromagnet slab of zero conductivity under the influence of an external magnetic field and submitted to an electromagnetic wave perturbation as described by coupled complex-valued system equations. As a result, with the aid of the Painlevé analysis, we find that for the above ferrite system to support microwave excitations, the damping factor raised from the Landau–Lifshitz–Gilbert formalism ought to vanish while implying the existence of some set of infinite law of conservations. Following the underlying feature of Kruskal's approach to generating in a straightforward manner some typical solutions, we henceforth discuss the physical implications of the traveling waveguide excitations. [ABSTRACT FROM AUTHOR]

The traveling wave tube (TWT) is a powerful vacuum electronic device used to amplify radio-frequency signals with numerous applications, including radar, television, and telephone satellite communications. TWT design in a nutshell comprises a pencil-like electron beam (e-beam) in vacuum interacting with a guiding slow-wave structure (SWS). In our studies here, the e-beam is represented by one-dimensional electron flow and SWS is represented by a transmission line (TL). The interaction between the e-beam and the TL is modeled by an analytic theory that generalizes the well-known Pierce model by taking into account the so-called space-charge effects, particularly electron-to-electron repulsion (debunching). Many important aspects of the analytic theory of TWTs have been already analyzed in our monograph on the subject. The focus of the studies here is on degeneracies of the TWT dispersion relations, particularly on exceptional points of degeneracy and their applications. The term exceptional point of degeneracy (EPD) refers to the property of the relevant matrix to have a nontrivial Jordan block structure. Using special parameterization, particularly suited to chosen EPD, we derive exact formulas for the relevant Jordan basis, including the eigenvectors and the so-called root vector associated with the Jordan block. Based on these studies, we develop a constructive approach to sensing of small signals. [ABSTRACT FROM AUTHOR]

The expansion of quantum states and operators in terms of Fock states plays a fundamental role in the field of continuous-variable quantum mechanics. In particular, for general single-mode Gaussian operators and Gaussian noisy states, many different approaches have been used in the evaluation of their Fock representation. In this paper, a natural approach has been applied using exclusively the operational properties of the Hermite and Hermite-like polynomials and showing their fundamental role in this field. Closed-form results in terms of polynomials, exponentials, and simple algebraic functions are the major contribution of the paper. [ABSTRACT FROM AUTHOR]

We investigate the most general form of the one-dimensional Dirac Hamiltonian HD in the presence of scalar and pseudoscalar potentials. To seek embedding of supersymmetry (SUSY) in it, as an alternative procedure to directly employing the intertwining relations, we construct a quasi-Hamiltonian K , defined as the square of HD, to explore the consequences. We show that the diagonal elements of K under a suitable approximation reflect the presence of a superpotential, thus proving a useful guide in unveiling the role of SUSY. For illustrative purposes, we apply our scheme to the transformed one-dimensional version of the planar electron Hamiltonian under the influence of a magnetic field. We generate spectral solutions for a class of isochronous potentials. [ABSTRACT FROM AUTHOR]

In this paper, an algebraic method to obtain the solution of linear partial differential equations of the evolution type is discussed. The proposed method exploits the Lie differential operators and their matrix realization, to reduce the equation to an easily solvable generalized matrix form. Some applications to problems of specific interest are also discussed. [ABSTRACT FROM AUTHOR]

All finite-dimensional irreducible representations of the general linear Lie superalgebra gl(2/2) are written down in matrix form. The basis within each representation space is chosen in such a way that it makes evident the decomposition of gl(2/2) into irreducible representations of its even subalgebra gl(2)+gl(2). Special attention is devoted to the analysis of all nontypical representations and some indecomposible representations. [ABSTRACT FROM AUTHOR]

In this paper the development of our previously published theory of approximations for the Chandler–Gibson (CG) equations is continued. In particular, our approximation theory is rigorously brought to the point where N-particle scattering calculations can begin. This is accomplished by mapping the CG operator equations into a function equation form, where the unknowns belong to a new (third!) computational Hilbert space L. This mapping is facilitated by rescaling the Jacobi momentum variables for the relative free motion of the asymptotic clusters so that surfaces of constant kinetic energy are hyperspheres. The input terms to the resulting equations are expanded in a basis on the surface of the kinetic energy hypersphere. This leads to a system of infinitely many coupled one-dimensional integral equations with the kinetic energy as the continuous variable. The half-on-shell variant of these equations is then transformed to a K-matrix form. Our approximations result from truncating this system to a finite number of equations, which is equivalent to using a finite basis approximation of the original input terms. The basis sets could be hyperspherical harmonics, but the use of hyperspherical spline functions is also proposed. Our method generalizes the well-known method of partial waves for channels with two clusters, and it accommodates breakup channels in a straightforward way. [ABSTRACT FROM AUTHOR]

Corrections and clarifications are made of some past treatments of the variational nature of the eigenfrequency calculation for dispersion equations and new results are presented. The main conclusions are the following: (1) Any relation between a normal mode and its dual must be consistent with the fact that the boundary conditions satisfied by the normal mode may differ from the adjoint boundary conditions satisfied by the dual. This will affect whether or not a given bilinear form will yield a variational result for the eigenfrequency. (2) If a dispersion matrix is constructed from the dispersion operator by using left and right basis functions that satisfy homogeneous boundary conditions on the dual eigenfunction and the eigenfunction, respectively, then generally a second-order accurate eigenfrequency is obtained by solving the matrix form of the dispersion equation. (3) When solving for the normal modes in terms of perturbation potentials, the adjoint boundary conditions are gauge dependent. For cases where the adjoint boundary conditions allow only the trivial solution for the dual eigenfunction, it may be possible to obtain variational results for the eigenfrequency by requiring that the trial functions for the normal mode and its dual satisfy variational boundary conditions. [ABSTRACT FROM AUTHOR]

In a series of two papers all finite-dimensional irreducible representations of the special linear Lie superalgebra sl(1,n) are written down in a matrix form. This paper develops a background for constructing the representations. Expressions for the transformation of the basis under the action of the generators are given for all induced and, hence, for all typical sl(1,n) modules. [ABSTRACT FROM AUTHOR]

A method for inverting observations on quantum mechanical systems to obtain estimates of unknown parameters residing in the Hamiltonian is presented. The quantal system is represented in matrix form with respect to a chosen basis, and it is assumed that the associated expansion coefficients are truncated to a finite dimension. The uncontrollable laboratory noise will be modeled by means of an inhomogeneous white noise process so that the experimental observations are represented as stochastic variables satisfying a stochastic differential equation. It will be assumed that measurements obtained from an experiment are now equivalent to a realization of these stochastic variables. It is known from filtering theory that the minimum variance estimate of the unknown parameters in the quantal model is now given by the expectation of the unknowns conditional on this realization. This estimator can be calculated analytically from the associated a posteriori probability density if the original quantal system does not contain any random elements. This probability density for the unknown matrix elements is calculated, and it is demonstrated that for a full Hamiltonian matrix the asymptotic variance of the parameter estimator decreases as a third power in time and a fourth power in the initial conditions. Some differences with the minimum least-square method are mentioned, and a few issues of numerical implementation are discussed. [ABSTRACT FROM AUTHOR]

A non-self-adjoint Sturmian eigenvalue equation of the form Av=f, encountered in quantum scattering theory, is solved as a complex general matrix eigenvalue problem. The matrix form is obtained on expansion of the solution in a discrete set of spherical Sturmian–Bessel functions of complex argument. This set of basis functions gives better convergence behavior for both the eigenvalues and eigenfunctions when compared to the results of a Chebyshev polynomial method reported previously. [ABSTRACT FROM AUTHOR]

Holstein and Primakoff derived long ago the boson realization of a su(2) Lie algebra for an arbitrary irreducible representation (irrep) of the SU(2) group. The corresponding result for su(1,1)≅sp(2) is also well known. This raises the question of whether it is possible to obtain in an explicit, analytic, and closed form, and for any integer d, the boson realization of a sp(2d) Lie algebra for an arbitrary irrep of the Sp(2d) group, which is a problem of considerable physical interest. The case d=2 already illustrates the problem in its full generality and thus in this paper we concentrate on sp(4). The Dyson realization is well known, and the passage to bosons satisfying the appropriate Hermiticity conditions can be done by a similarity transformation through an operator K. What we want, though, is an explicit boson realization for sp(2d) similar to the one that exists for sp(2). In Sec. VI we show how we can get it for sp(4) if the operator K is known. Unfortunately while the matrix form of K2 can be explicitly derived from definite recursion relations, the same cannot be said of K as it involves, in general, the solution of algebraic equations of high degree. Thus the conclusion, corroborated also by a classical analysis where K does not appear, is that an explicit, analytic, and closed boson realization of sp(4), and thus also of sp(2d), is only possible for particular irreps of the corresponding groups. [ABSTRACT FROM AUTHOR]

We consider a problem of separation of variables for the Lax-integrable Hamiltonian systems governed by gl(n) ⊗ gl(n)-valued classical r-matrices r(u, v). We report on a class of classical non-skew-symmetric non-dynamical gl(n) ⊗ gl(n)-valued r-matrices rJ(u, v) labeled by arbitrary anisoropy matrix J ∈ gl(n) for which the "magic recipe" of Sklyanin [Prog. Theor. Phys., 118, 35 (1995)] in the theory of variable separation is applicable. An example of n = 3 corresponding to gl(3) ⊗ gl(3)-valued r-matrices is elaborated in detail. For the case of the r-matrices rJ(u, v) and n = 3, the coordinates of separation, the reconstruction formulas, and the Abel-type equations are explicitly written for the different types of matrices J. [ABSTRACT FROM AUTHOR]

A family of non-Hermitian real and tridiagonal-matrix candidates H (N) (λ) = H 0 (N) + λ W (N) (λ) for a hiddenly Hermitian (a.k.a. quasi-Hermitian) quantum Hamiltonian is proposed and studied. Fairly weak assumptions are imposed upon the unperturbed matrix [the square-well-simulating spectrum of H 0 (N) is not assumed equidistant)] and upon its maximally non-Hermitian N-parametric antisymmetric-matrix perturbations [matrix W(N)(λ) is not even required to be P T -symmetric]. Despite that, the "physical" parametric domain D [ N ] is (constructively) shown to exist, guaranteeing that in its interior, the spectrum remains real and non-degenerate, rendering the quantum evolution unitary. Among the non-Hermitian degeneracies occurring at the boundary ∂ D [ N ] of the domain of stability, our main attention is paid to their extreme version corresponding to Kato's exceptional point of order N (EPN). The localization of the EPNs and, in their vicinity, of the quantum-phase-transition boundaries ∂ D [ N ] is found feasible, at the not too large N, using computer-assisted symbolic manipulations, including, in particular, the Gröbner-basis elimination and the high-precision arithmetics. [ABSTRACT FROM AUTHOR]

We consider polygon and simplex equations, of which the simplest nontrivial examples are pentagon (5-gon) and Yang–Baxter (2-simplex), respectively. We examine the general structure of (2n + 1)-gon and 2n-simplex equations in direct sums of vector spaces. Then, we provide a construction for their solutions, parameterized by elements of the Grassmannian Gr(n + 1, 2n + 1). [ABSTRACT FROM AUTHOR]

The discrete orthogonality relations hold for all the orthogonal polynomials obeying three term recurrence relations. We show that they also hold for multi-indexed Laguerre and Jacobi polynomials, which are new orthogonal polynomials obtained by deforming these classical orthogonal polynomials. The discrete orthogonality relations could be considered as a more encompassing characterization of orthogonal polynomials than the three term recurrence relations. As the multi-indexed orthogonal polynomials start at a positive degree ℓ D ≥ 1 , the three term recurrence relations are broken. The extra ℓ D "lower degree polynomials," which are necessary for the discrete orthogonality relations, are identified. The corresponding Christoffel numbers are determined. The main results are obtained by the blow-up analysis of the second order differential operators governing the multi-indexed orthogonal polynomials around the zeros of these polynomials at a degree ℓ D + N. The discrete orthogonality relations are shown to hold for another group of "new" orthogonal polynomials called Krein–Adler polynomials based on the Hermite, Laguerre, and Jacobi polynomials. [ABSTRACT FROM AUTHOR]

We consider the problem of the propagation of an electric field generated by periodic pumping in a stable medium of two-level atoms as the mixed problem for the Maxwell–Bloch equations without spectrum broadening. An approach to the study of such a problem is proposed. We use the inverse scattering transform method in the form of the matrix Riemann–Hilbert (RH) problem, using simultaneous spectral analysis of both the Lax equations. The proposed matrix RH problem solves the problem of the propagation of a sinusoidal signal in an unperturbed stable medium (attenuator). It is proved that this RH problem provides the causality principle for the region t < x, and for the region of the light cone, 0 < x < t allows us to find the asymptotics of the transmitted signal. First, we study the asymptotics of the RH problem for large times, and then, we obtain asymptotic formulas for the mixed problem solution of the Maxwell–Bloch equations when the attenuator is long enough. Three sectors are obtained in the light cone where the asymptotics have essentially different behaviors. [ABSTRACT FROM AUTHOR]

It is shown that the zilch conservation law arises as the Noether current corresponding to a variational symmetry of a duality-symmetric Maxwell Lagrangian. The action of the corresponding symmetry generator on the duality-symmetric Lagrangian, while non-vanishing, is a total divergence as required by the Noether theory. The variational nature of the zilch conservation law was previously known only for some of the components of the zilch tensor, notably the optical chirality. By contrast, our analysis is fully covariant and is, therefore, valid for all components of the zilch tensor. The analysis is presented here for both the real and complex versions of duality-symmetric Maxwell Lagrangians. [ABSTRACT FROM AUTHOR]

We advance here an algorithm of the synthesis of lossless electric circuits such that their evolution matrices have the prescribed Jordan canonical forms subject to natural constraints. Every synthesized circuit consists of a chain-like sequence of LC-loops coupled by gyrators. All involved capacitances, inductances, and gyrator resistances are either positive or negative with values determined by explicit formulas. A circuit must have at least one negative capacitance or inductance for having a nontrivial Jordan block for the relevant matrix. [ABSTRACT FROM AUTHOR]

We present exact solutions for the Lippmann–Schwinger equation in two dimensions for circular boundary walls in three cases: (i) a finite number N of concentric barriers; (ii) a single barrier with Dirac delta derivatives, in the sense of distribution theory, namely, angular, normal, and along the curve; and (iii) a single barrier with an arbitrary distribution. As an application of this last result, we obtain conditions that must be fulfilled in order for the barrier to become invisible. [ABSTRACT FROM AUTHOR]

In this paper, continuous-time master equations with finite states employed in nonequilibrium statistical mechanics are formulated in the language of discrete geometry. In this formulation, chains in algebraic topology are used, and master equations are described on graphs that consist of vertices representing states and of directed edges representing transition matrices. It is then shown that master equations under the detailed balance conditions are equivalent to discrete diffusion equations, where the Laplacians are defined as self-adjoint operators with respect to introduced inner products. An isospectral property of these Laplacians is shown for non-zero eigenvalues, and its applications are given. The convergence to the equilibrium state is shown by analyzing this class of diffusion equations. In addition, a systematic way to derive closed dynamical systems for expectation values is given. For the case that the detailed balance conditions are not imposed, master equations are expressed as a form of a continuity equation. [ABSTRACT FROM AUTHOR]

Separated variables for a classical GL(3) magnetic chain are coordinates of a generic positive divisor D of degree n on a genus g nonhyperelliptic algebraic curve. Because n > g, this divisor D has unique representative ρ(D) in the Jacobian, which can be constructed by using dim|D| = n − g steps of Abel’s algorithm. We study the properties of the corresponding chain of divisors and prove that the classical GL(3) magnetic chain is a superintegrable system with dim|D| = 2 superintegrable Hamiltonians. [ABSTRACT FROM AUTHOR]

States of Low Energy (SLEs) are exact Hadamard states defined on arbitrary Friedmann–Lemaître spacetimes. They are constructed from a fiducial state by minimizing the Hamiltonian's expectation value after averaging with a temporal window function. We show the SLE to be expressible solely in terms of the (state independent) commutator function. They also admit a convergent series expansion in powers of the spatial momentum, both for massive and for massless theories. In the massless case, the leading infrared behavior is found to be Minkowski-like for all scale factors. This provides a new cure for the infrared divergences in Friedmann–Lemaître spacetimes with accelerated expansion. As a consequence, massless SLEs are viable candidates for pre-inflationary vacua, and in a soluble model, they are shown to entail a qualitatively correct primordial power spectrum. [ABSTRACT FROM AUTHOR]

We find and classify all bialgebras and Hopf algebras or "quantum groups" of dimension ≤4 over the field F 2 = { 0 , 1 }. We summarize our results as a quiver, where the vertices are the inequivalent algebras and there is an arrow for each inequivalent bialgebra or Hopf algebra built from the algebra at the source of the arrow and the dual of the algebra at the target of the arrow. There are 314 distinct bialgebras and, among them, 25 Hopf algebras, with at most one of these from one vertex to another. We find a unique smallest noncommutative and noncocommutative one, which is moreover self-dual and resembles a digital version of uq(sl2). We also find a unique self-dual Hopf algebra in one anyonic variable x4 = 0. For all our Hopf algebras, we determine the integral and associated Fourier transform operator, viewed as a representation of the quiver. We also find all quasitriangular or "universal R-matrix" structures on our Hopf algebras. These induce solutions of the Yang–Baxter or braid relations in any representation. [ABSTRACT FROM AUTHOR]

We study an infinite one-dimensional Ising spin chain where each particle interacts only with its nearest neighbors and is in contact with a heat bath with temperature decaying hyperbolically along the chain. The time evolution of the magnetization (spin expectation value) is governed by a semi-infinite Jacobi matrix. The matrix belongs to a three-parameter family of Jacobi matrices whose spectral problem turns out to be solvable in terms of the basic hypergeometric series. As a consequence, we deduce the essential properties of the corresponding orthogonal polynomials, which seem to be new. Finally, we return to the Ising model and study the time evolution of magnetization and two-spin correlations. [ABSTRACT FROM AUTHOR]

This article constructs the Hilbert space for the algebra αβ − eiθβα = 1 that provides a continuous interpolation between the Clifford and Heisenberg algebras—this particular form is inspired by the properties of anyons. We study the eigenvalues of a generalized number operator (N = β α) and construct the Hilbert space, classified by values of a complex coordinate (λ0): the eigenvalues lie on a circle. For θ being an irrational multiple of 2π, we get an infinite-dimensional representation; however, for a rational multiple ( M N ) of 2π, it is finite-dimensional, parameterized by the complex coordinate λ0. The case for N = 2, θ = π is the usual Clifford algebra for fermions, while the case for N = ∞, θ = 0 is the Heisenberg algebra of bosons, albeit with two copies for positive and negative eigenvalues. We find a smooth transition from the fermion to the boson situation as N → ∞ from N = 2. After constructing the Hilbert space from the algebra, the cases for N = 2, 3 can be mapped to the "fuzzy sphere" of SU(2), while for general N, the "fuzzy pancake" is found to be the correct representation. Then, we motivate the study of coherent states, which are the eigenstates of α (the lowering operator), labeled by complex numbers for non-zero λ0. We specialize the study of coherent states to the very interesting case of λ0 = 0 and construct a calculus of the generalized Grassmann variables that result, applying it to compute a partition function for these particles. We then make some remarks about extending this study to that of anyons. [ABSTRACT FROM AUTHOR]

We study the scattering theory for the Schrödinger and wave equations with rough potentials in a scale of homogeneous Sobolev spaces. The first half of this paper is concerned with an inverse-square potential in both of subcritical and critical constant cases, which is a particular model of scaling-critical singular perturbations. In the subcritical case, the existence of the wave and inverse wave operators defined on a range of homogeneous Sobolev spaces is obtained. In particular, we have the scattering to a free solution in the homogeneous energy space for both of the Schrödinger and wave equations. In the critical case, it is shown that the solution is asymptotically a sum of an n-dimensional free wave and a rescaled two-dimensional free wave. The second half of this paper is concerned with a generalization to a class of strongly singular decaying potentials. We provide a simple criterion in an abstract framework to deduce the existence of wave operators defined on a homogeneous Sobolev space from the existence of the standard ones defined on a base Hilbert space. [ABSTRACT FROM AUTHOR]

We study the classical and semiclassical time evolutions of subsystems of a Hamiltonian system; this is done using a generalization of Heller's thawed Gaussian approximation introduced by Littlejohn. The key tool in our study is an extension of Gromov's "principle of the symplectic camel" obtained in collaboration with Dias, de Gosson, and Prata [arXiv:1911.03763v1 [math.SG] (2019)]. This extension says that the orthogonal projection of a symplectic phase space ball on a phase space with a smaller dimension also contains a symplectic ball with the same radius. In the quantum case, the radii of these symplectic balls are taken equal to ℏ and represent the ellipsoids of minimum uncertainty, which we called "quantum blobs" in previous work. [ABSTRACT FROM AUTHOR]

We consider the problem of variable separation for the classical integrable Hamiltonian systems possessing gl(n)-valued Lax matrices satisfying quadratic Poisson brackets of Freidel and Maillet [Phys. Lett. B 262(2-3), 278 (1991)] with spectral-parameter dependent a-b-c-d tensors. We formulate, in terms of the corresponding a-b-c-d tensors, a sufficient condition that guarantees that the separating functions A(u), B(u) of Sklyanin [Commun. Math. Phys. 150(1), 181 (1992)], Scott [e-print arXiv:hep-th 940303 (1994)], and Gekhtman [Commun. Math. Phys. 167, 593 (1995)] produce canonical coordinates. We consider an important subcase of a-b-c-d algebras, namely, the case of classical reflection equation algebras [E. Sklyanin, J. Phys. A: Math. Gen. 21(10), 2357 (1988)], and formulate, in terms of the corresponding r-s-matrices, the analogous sufficient condition that guarantees the canonicity of the constructed coordinates. For the case s = 0, we recover our previous results on the variable separation for quadratic Sklyanin brackets [B. Dubrovin and T. Skrypnyk, J. Math. Phys. 60, 093506 (2019)]. We consider an example of gl(n) ⊗ gl(n)-valued trigonometric a-b-c-d tensors that satisfy the considered condition and find a class of Lax matrices for them for which the obtained set of canonical coordinates is complete. [ABSTRACT FROM AUTHOR]

The relevance in physics of non-Hermitian operators with real eigenvalues is being widely recognized not only in quantum mechanics but also in other areas, such as quantum optics, quantum fluid dynamics, and quantum field theory. In this note, a quantum system described by a non-Hermitian Hamiltonian, which is constituted by two types of interacting bosons, is investigated. The real eigenvalues of the Hamiltonian are explicitly determined, as well as complete biorthogonal sets of eigenfunctions of the Hamiltonian and its adjoint. The diagonal representation of H is obtained using pseudo-bosonic operators. [ABSTRACT FROM AUTHOR]

Motivated by the realization of the Yang–Baxter equation of 2D integrable models in the 4D gauge theory of Costello–Witten–Yamazaki (CWY), we study the embedding of integrable 2D Toda field models inside this construction. This is done by using the Lax formulation of 2D integrable systems and by thinking of the standard Lax pair L± in terms of components of CWY gauge connection, propagating along particular directions in the gauge bundle. We also use the results of the CWY theory to build quantum line operators for 2D Toda models and compute the one loop contribution of two intersecting lines exchanging one gluon. Other features such as local symmetries and comments on extension of our method to other 2D integrable models are also discussed. We also comment on some basic points that still need a refinement before discussing about a fully consistent embedding of Lax pairs into CWY theory. [ABSTRACT FROM AUTHOR]

In this article, we answer the following question: If the wave equation possesses bound states, but it is exactly solvable for only a single non-zero energy, can we find all bound state solutions (energy spectrum and associated wavefunctions)? To answer this question, we use the "tridiagonal representation approach" to solve the wave equation at the given energy by expanding the wavefunction in a series of energy-dependent square integrable basis functions in configuration space. The expansion coefficients satisfy a three-term recursion relation, which is solved in terms of orthogonal polynomials. Depending on the selected energy, we show that one of the potential parameters must assume a value from within a discrete set called the "potential parameter spectrum" (PPS). This discrete set is obtained from the spectrum of the above polynomials and can be either a finite or an infinite set. Inverting the relation between the energy and the PPS gives the bound state energy spectrum. Therefore, the answer to the above question is affirmative. [ABSTRACT FROM AUTHOR]

In this paper, we show that it is always possible to deform a differential equation ∂xΨ(x) = L(x)Ψ(x) with L (x) ∈ s l 2 (C) (x) by introducing a small formal parameter ℏ in such a way that it satisfies the topological type properties of Bergère, Borot, and Eynard [Annales Henri Poincaré 16(12), 2713–2782 (2015)]. This is obtained by including the former differential equation in an isomonodromic system and using some homogeneity conditions to introduce ℏ. The topological recursion is then proved to provide a formal series expansion of the corresponding tau-function whose coefficients can thus be expressed in terms of intersections of tautological classes in the Deligne–Mumford compactification of the moduli space of surfaces. We present a few examples including any Fuchsian system of s l 2 (C) (x) as well as some elements of Painlevé hierarchies. [ABSTRACT FROM AUTHOR]

A one-dimensional interacting particle system is revisited. It has discrete time, and its components are located in the set of integers. These components can disappear in the functioning process. Each component assumes two possible states, called plus and minus, and interacts at every time step only with its nearest neighbors. The following two transformations happen: The first one is called flip, under its action, a component in state minus turns into a plus with probability β. The second one is called annihilation, under its action, whenever a component in state plus is a left neighbor of a component in state minus, both components disappear with probability α. Let us consider a set of initial measures to the process. For these measures, we show the upper bound for the mean time of convergence, which is a function of the initial measure. Moreover, we obtain the upper bound to the mean quantity of minuses on the process in each time step. Considering the initial measure concentrated at the configuration whose components are in the state minus, we improved a well-known result that the process is non-ergodic when β < α2/250. Now, we are able to offer non-ergodicity when β < 9α2/1000. We also established new conditions to the ergodicity of the process. Finally, we performed some Monte Carlo simulations for this process. [ABSTRACT FROM AUTHOR]

Methods elaborated in quantum mechanics for the Landau–Zener problem are generalized to study the non-adiabatic transitions in a wide class of problems of wave propagation, in particular in the waveguide problems. If the properties of the waveguide slowly vary along its axis and the phase velocities of two modes have a degeneracy point or are almost degenerate near some point, the transformation of modes may occur. The conditions are formulated under which we can find formal asymptotic expansions of modes outside the vicinity of the degeneracy point and write out explicitly the transition matrix. The starting point is rewriting the governing equations in the form of the Schrödinger type equation. The Hamiltonian is assumed to be the result of a small perturbation of an operator with a degeneracy point of the crossing types of two eigenvalues. The perturbation of the Hamiltonian produces a close pair of simple degeneracy points. Two regimes of mode transformation for the Schrödinger type equation are identified: avoided crossing of eigenvalues (corresponding to complex degeneracy points) and an explicit unavoidable crossing (with real degeneracy points). [ABSTRACT FROM AUTHOR]

We consider trigonometric solutions of the Kadomtsev–Petviashvili (KP) hierarchy. It is known that their poles move as particles of the Calogero–Moser model with trigonometric potential. We show that this correspondence can be extended to the level of hierarchies: the evolution of the poles with respect to the kth hierarchical time of the KP hierarchy is governed by a Hamiltonian that is a linear combination of the first k higher Hamiltonians of the trigonometric Calogero–Moser hierarchy. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the problem of imploding shock waves in dusty real reacting gases with the same γ-law via the Lie group invariance method. Necessary invariance group properties of an ambient gas for strong shocks are presented and yield the forms of the reaction rate in different cases. The collapse of imploding shocks for a radially symmetric flow of non-ideal dusty reacting gases is completely characterized. The effects of mass consolidation of dust particles, relative specific heat, van der Waals excluded volume, energies of formation of mixtures, and the ratio of the density of dust particles to that of density of the gas on shock involution are obtained. [ABSTRACT FROM AUTHOR]

The asymptotic iteration method is a technique for solving analytically and approximately the linear second-order differential equation, especially the eigenvalue problems that frequently appear in theoretical and mathematical physics. The analysis and mathematical justifications of the success and failure of the asymptotic iteration method are detailed in this work. A theorem explaining why the asymptotic iteration method works for the eigenvalue problem is presented. As a byproduct, a new procedure to generate unlimited classes of exactly solvable differential equations is also introduced. [ABSTRACT FROM AUTHOR]

We study the time evolution of two coupled quantum harmonic oscillators interacting through nonlinear optomechanical-like Hamiltonians that include cross-Kerr interactions. We employ techniques developed to decouple the time-evolution operator and obtain the analytical solution for the time evolution of the system. We apply these results to obtain explicit expressions of a few quantities of interest. Our results do not require approximations and therefore allow us to study the nature and implications of the full nonlinearity of the system. As a potential application, we show that it is possible to greatly increase the population of phonons using a suitable combination of cubic and cross-Kerr interactions. [ABSTRACT FROM AUTHOR]

We study a non-Hermitian chiral random matrix of which the eigenvalues are complex random variables. The empirical distributions and the radius of the eigenvalues are investigated. The limit of the empirical distributions is a new probability distribution defined on the complex plane. The graphs of the density functions are plotted; the surfaces formed by the density functions are understood through their convexity and their Gaussian curvatures. The limit of the radius is a Gumbel distribution. The main observation is that the joint density function of the eigenvalues of the chiral ensemble, after a transformation, becomes a rotation-invariant determinantal point process on the complex plane. Then, the eigenvalues are studied by the tools developed by Jiang and Qi [J. Theor. Probab. 30, 326 (2017); 32, 353 (2019)]. Most efforts are devoted to deriving the central limit theorems for distributions defined by the Bessel functions via the method of steepest descent and the estimates of the zero of a non-trivial equation as the saddle point. [ABSTRACT FROM AUTHOR]