Russian Journal of Mathematical Physics最新文献

In this paper, we define the probabilistic degenerate Laguerre polynomials associated with random variables and the probabilistic degenerate generalized Laguerre polynomials associated with random variables. We investigate some expressions, recurrence relations, and properties associated with the probabilistic degenerate Laguerre polynomials, the probabilistic degenerate generalized Laguerre polynomials, the probabilistic Lah numbers, and the partial Bell polynomials.

DOI 10.1134/S1061920824040095

Uniform asymptotic solutions of the field structures in the vicinity of focusing based on the use of the Maslov canonical operator leads to investigation of special functions of wave catastrophe (SWC) and their first derivatives. The method for constructing a system of differential equations to determine the fundamental vector of special functions of wave catastrophes (SWC) is created. This approach allows us to reduce the solution of the problem of determining the SWCs and their derivatives to the solution of the Cauchy problem for a system of ordinary differential equations. The paper provides examples of the construction of such systems for special functions of edge catastrophes corresponding to Lagrange manifolds with boundary and special functions of main catastrophes corresponding to Lagrange manifolds without restrictions.

DOI 10.1134/S1061920824040083

The logistic equation with delay and diffusion, which is important in mathematical ecology, is considered. It is assumed that the boundary conditions at either end of the interval [0,1] contain parameters. The problem of local dynamics, in a neighborhood of the equilibrium state, of the corresponding boundary value problem is investigated for all values of the boundary condition parameters. Critical cases are identified in the problem of stability of the equilibrium state and normal forms are constructed, which are scalar complex ordinary differential equations of the first order. Their nonlocal dynamics determines the behavior of solutions of the original problem in a small neighborhood of the equilibrium state. The problem of the role of asymptotically small values of the diffusion coefficient in the dynamics of the boundary value problems under consideration is studied separately. In particular, it is shown that boundary layer functions may arise when constructing asymptotic solutions in a neighborhood of the boundary points 0 and 1.

DOI 10.1134/S106192082404006X

The paper is devoted to constructing the global asymptotics of Jacobi polynomials by the method of “real semiclassics for problems with complex phases,тАЩтАЩ which is based on the study of recurrence relations. The method is based on the semiclassical approximation and the study of the geometry and types of singularities of the arising Lagrangian manifolds. While manifolds with a turning point in whose neighborhood the asymptotics is determined by the Airy function are well studied, the methods for the case in which the asymptotics is determined by the Bessel functions are not so well developed. In this paper, we demonstrate the application of the above-mentioned method in both situations, in particular, we describe the Lagrangian manifold that arises in the second case.

DOI 10.1134/S1061920824040162

Using Maslov’s canonical operator in the Cauchy problem for a Dirac equation, we consider the asymptotics of the solution of the Cauchy problem in which the potential depends irregularly on a small parameter.

DOI 10.1134/S1061920824040010

We consider a Hamiltonian system consisting of the Klein–Gordon field coupled to an infinite harmonic crystal. The dynamics of the coupled system is invariant with respect to the space translations in (mathbb{Z}^d), (dge1). We study the Cauchy problem and assume that the initial date is a random function. We introduce the family of initial probability measures ({mu_0^varepsilon,varepsilon >0}) depending on a small parameter (varepsilon) and slowly varying on the linear scale (1/varepsilon). For times of order (varepsilon^{-kappa}), (kappa>0), we study the asymptotics of the distributions of the random solution as (varepsilonto0). In particular, we show that, for (kappa=1) and (kappa=2), the limiting covariance is governed by the hydrodynamic equations of the Euler and Navier–Stokes type, respectively.

DOI 10.1134/S1061920824040034

We consider a certain class of infinite continued fractions such that their elements are bounded operators in a Hilbert space. They can be regarded as analogs of (J)-fractions related to the classical moment problem and the theory of Jacobi operators. To each of these operator (J)-fractions there corresponds a band operator generated by three-diagonal infinite matrix which entries coincide with the elements of this continued fraction. Using the theory of such band operators, we establish the basic properties of the continued fractions under consideration: their expansion algorithm, a criterion for existence of this expansion, and the uniqueness theorem. Also we establish the convergence (at a geometric rate) of an operator (J)-fraction outside the numerical range of the corresponding band operator to the Weyl function of the latter. We show how these results can be applied for solving quadratic operator equations.

DOI 10.1134/S1061920824040125

We study a one-dimensional quasiperiodic difference Schrödinger operator with a potential obtained by restricting a certain meromorphic function to the integer lattice. Assuming that the coupling constant is sufficiently small, we asymptotically describe a series of intervals contained in spectral gaps, their centers, and lengths. The lengths of these intervals decrease exponentially as their number increases, and the rate of their decrease is determined by the distance from the poles of the potential to the real axis.

DOI 10.1134/S1061920824040046

The equation (- Delta u + V u = 0) in the cylinder (mathbb{R} times (0,2pi)^d) with periodic boundary conditions is considered. The potential (V) is assumed to be bounded, and both functions (u) and (V) are assumed to be real-valued. It is shown that the fastest rate of decay at infinity of nontrivial solution (u) is (Oleft(e^{-c|w|}right)) for (d=1) or (2), and (Oleft(e^{-c|w|^{4/3}}right)) for (dge 3). Here (w) stands for the axial variable.

DOI 10.1134/S1061920824040058

In this paper, we revisit McLaughlin’s inverse problem, which consists in the recovery of the fourth-order differential operator from the eigenvalues and two sequences of norming constants. We prove the uniqueness for solution of this problem for the first time. Moreover, we obtain an interpretation of McLaughlin’s problem in the framework of the general inverse problem theory by Yurko for differential operators of arbitrary orders. An advantage of our approach is that it requires neither the smoothness of the coefficients nor the self-adjointness of the operator. In addition, we establish the connection between McLaughlin’s problem and Barcilon’s three-spectra inverse problem.

DOI 10.1134/S1061920824040022