Russian Journal of Mathematical Physics最新文献

The paper deals with a Hamiltonian, namely, with a semi-bounded self-adjoint operator that is attributed to the problem of scattering of three one-dimensional particles with point interaction in pairs, in other words, with ( delta )-functional singular potential of interaction. The support of the potential in the Hamiltonian coincides with a symmetric star-graph having six leads on the two-dimensional plane. Due to the symmetry, we find that such a model is exactly solvable, which means that the eigenfunctions of the discrete spectrum and the generalized eigenfunctions of the essential (absolutely continuous) spectrum are determined explicitly, i.e., by quadrature. In this (first part) of our work we describe the discrete spectrum and the eigenfunctions.

DOI 10.1134/S1061920825600850

The paper presents two incongruent flat simply connected domains with a visually smooth boundary for which the eigenvalues of the Dirichlet–Neumann operator, obtained using the algorithm cited in the paper, are practically identical.

DOI 10.1134/S1061920825601107

This paper introduces a novel generalization of Stirling and Lah numbers, termed “heterogeneous Stirling numbers,” which smoothly interpolate between these classical combinatorial sequences. Specifically, we define heterogeneous Stirling numbers of the second and first kinds, demonstrating their convergence to standard Stirling numbers as (lambda rightarrow 0) and to (signed) Lah numbers as (lambda rightarrow 1). We derive fundamental properties, including generating functions, explicit formulas, and recurrence relations. Furthermore, we extend these concepts to heterogeneous Bell polynomials, obtaining analogous results such as generating function, combinatorial identity and Dobinski-like formula. Finally, we introduce and analyse heterogeneous (r)-Stirling numbers of the second kind and their associated (r)-Bell polynomials.

DOI 10.1134/S1061920825601065

We study weakly compact subsets of normed linear spaces admitting, for each (varepsilon>0), an nw-continuous (varepsilon)-selection and such that the closure of their convex hull is weakly compact in this space. Such sets are shown to be convex. An application of this result to the linear manifold of all analytic functions in (L_1) is given.

DOI 10.1134/S1061920825600539

The stress-strain state arising after a two-stage spherically symmetric process “radial loading– partial unloading” in a hollow sphere (thick-walled spherical layer) nonuniform in radius is investigated. The inner boundary is fixed, and a radial compressive stress is applied to the outer boundary. The material of the sphere agrees with Ilyushin’s theory of small elastoplastic deformations. Both discontinuous and continuously stratified nonuniformity are admitted, while the bulk modulus of compression at all points of the material is assumed to be constant. At the first (intermediate) step, an excessive compression (over-compression) occurs, as a result of which the radius of the outer boundary becomes less than the required one. At the second step, where partial unloading occurs, the system is transferred to the required final position. Such a two-stage nature enables one to control the stress-strain state in the hollow sphere both as a whole and concerning its individual parameters.

DOI 10.1134/S1061920825030070

A special Cauchy problem for the Schrödinger equation with a delta potential localized on a half-line is addressed. From the viewpoint of high-frequency parabolic-equation heuristics, the problem could be viewed as an approximation to a paraxial (i.e., nearly tangential) diffraction of a plane wave incident on a screen. Explicit solution of the problem, found with the help of integral transformations, is subjected to exhaustive asymptotic investigation for all values of the complex coefficient of the potential. The asymptotic findings are qualitatively interpreted using diffraction terminology and quantitatively compared with the results of diffraction theory. Some effects that have no analogs in the related diffraction problems are noted. The solution is shown to be, in a certain range of parameters, an asymptotic solution of the Helmholtz equation with a delta potential.

DOI 10.1134/S1061920825600679

The theory of tetragonal curves is applied to the study of discrete integrable systems. Based on the discrete Lenard equation, we derive a hierarchy of Blaszak–Marciniak four-field lattice equations associated with the (4times4) matrix spectral problem. Resorting to the characteristic polynomial of the Lax matrix for the hierarchy of Blaszak–Marciniak four-field lattice equation, we introduce a tetragonal curve, a Baker-Akhiezer function, and three meromorphic functions on it. We study algebro-geometric properties of the tetragonal curve and asymptotic behaviors of the Baker-Akhiezer function and meromorphic functions near two infinite points. The straightening out of various flows is exactly given by means of the Abel map and the meromorphic differential. We finally obtain Riemann theta function solutions of the entire Blaszak–Marciniak four-field lattice hierarchy.

DOI 10.1134/S1061920825600667

The aim of this paper is to disprove the conjecture that the only Lie algebras admitting semisimple algebraic Nijenhuis operators are Abelian ones. We do not only provide examples in arbitrary dimension but also classify three-dimensional Lie algebras which admit such an operator in both complex and real cases. As a byproduct, we obtain a list of three-dimensional Lie groups which are left-invariant Nijenhuis manifolds.

DOI 10.1134/S1061920825600758

Given coefficient conditions of analytic in unit disk (mathbf{D}), we obtain the radius of starlike of order (alpha) and convex of order (alpha) for Cesàro operator of analytic functions (f) in (mathbf{D}). Furthermore, we consider the Cesàro operator of harmonic functions and give the radius of starlike of order (alpha) and convex of order (alpha) for Cesàro operator of harmonic functions (f) in (mathbf{D}). All results are sharp.

DOI 10.1134/S1061920824601484

In this paper, we are concerned with the asymptotic behavior and the Liouville theorem for the nonlinear integral equation

where (u in C^{infty}(mathbb{R}^{n},mathbb{S}^{k-1})) with (n geq 3), (k geq 2), (alpha in (0,n/2)), (mathbb{S}^{k-1}={u=(u_1,u_2,cdots,u_k) in mathbb{R}^{k}; |u|=1}), (e_k=(0,cdots,0,1)), (C_*) is a positive constant, and (ell in mathbb{R}^{k}) is a constant vector. We prove that, if (u_k in L^2(mathbb{R}^n)) and (nabla u in L^2(mathbb{R}^n) cap L^infty(mathbb{R}^n)), then (u to ell) as (|x| to infty) with (ell_k=0). Moreover, if (alpha in (1,n/2)), then (u equiv ell) on (mathbb{R}^n).

DOI 10.1134/S1061920825600576