Russian Journal of Mathematical Physics最新文献

The semiclassical asymptotics of the solution of the Cauchy problem for the Schrödinger equation with a delta potential localized on a surface of codimension 1 is described. The Schrödinger operator with a delta potential is defined using extension theory and specified by boundary conditions on this surface. The initial conditions are chosen in the form of a narrow peak, which is a Gaussian packet, localized in a small neighborhood of a surface of arbitrary dimension, and oscillating rapidly along it. The Maslov complex germ method is used to construct the asymptotics. The reflection of an isotropic manifold with a complex germ interacting with the delta potential is described.

DOI 10.1134/S1061920824030142

The coincidence of the ( operatorname{Ind} ) and (dim) dimensions for the first countable paracompact (sigma)-spaces is proved. This gives a positive answer to A.V. Arkhangel’skii’s question of whether the dimensions ( operatorname{ind} X), ( operatorname{Ind} X), and (dim X) are equal for the first countable spaces with a countable network.

DOI 10.1134/S1061920824030178

Let (Y) be a random variable whose moment generating function exists in a neighborhood of the origin. The aim of this paper is to study probabilistic Bernoulli polynomials of order (r) associated with (Y) and probabilistic multi-poly-Bernoulli polynomials associated with (Y). They are respectively probabilistic extensions of Bernoulli polynomials of order (r) and multi-poly-Bernoulli polynomials. We find explicit expressions, certain related identities and some properties for them. In addition, we treat the special cases of Poisson, gamma and Bernoulli random variables.

DOI 10.1134/S1061920824030087

This paper studies the uniqueness of a weak solution of the incompressible Navier–Stokes Equations in the 3-dimensional case. Here the investigation is provided by using two different approaches. The first (the main) result is obtained for given functions possessing a certain smoothness, using a new approach. The other result works without additional conditions but is, in some sense, a “local” result, investigated by another approach. In addition, here the solvability and uniqueness of weak solutions to the auxiliary problems derived from the main problem are investigated.

DOI 10.1134/S1061920824030154

We study asymptotic spectral properties of the Bochner–Schrödinger operator (H_{p}=frac 1pDelta^{L^potimes E}+V) on high tensor powers of a Hermitian line bundle (L) twisted by a Hermitian vector bundle (E) on a Riemannian manifold (X) of bounded geometry under the assumption that the curvature form of (L) is nondegenerate. At an arbitrary point (x_0) of (X), the operator (H_p) can be approximated by a model operator (mathcal H^{(x_0)}), which is a Schrödinger operator with constant magnetic field. For large (p), the spectrum of (H_p) asymptotically coincides, up to order (p^{-1/4}), with the union of the spectra of the model operators (mathcal H^{(x_0)}) over (X). We show that, if the union of the spectra of (mathcal H^{(x_0)}) over the complement of a compact subset of (X) has a gap, then the spectrum of (H_{p}) in the gap is discrete, and the corresponding eigensections decay exponentially away from a compact subset.

DOI 10.1134/S1061920824030099

We obtain a uniform decomposition into Casimir eigenspaces (most of which are irreducible) of the fourth power of the adjoint representation (mathfrak{g}^{otimes 4}) for all simple Lie algebras. We present universal, in Vogel’s sense, formulas for the dimensions and split Casimir operator’s eigenvalues of all terms in this decomposition. We assume that a similar uniform decomposition into Casimir eigenspaces with universal dimension formulas exists for an arbitrary power of the adjoint representations.

DOI 10.1134/S1061920824030038

Some necessary and sufficient conditions for the strong continuity of representations of topological groups in reflexive Fréchet spaces are obtained.

DOI 10.1134/S106192082403018X

This paper studies time-periodic solutions of singularly perturbed Tikhonov systems of reaction–diffusion–advection equations with nonlinearities that include the square of the gradient of the unknown function (KPZ nonlinearities). The boundary layer asymptotics of solutions are constructed for Neumann and Dirichlet boundary conditions. The study considers both the case of quasimonotone sources and systems without the quasimonotonicity condition. The asymptotic method of differential inequalities is used to prove theorems on the existence of solutions and their Lyapunov asymptotic stability.

DOI 10.1134/S1061920824030129

We construct an appropriate restriction of the 2-dimensional Laplace operator that has compact preresolvent though the resolvent of its Friedrichs extension is not compact and, moreover, its spectrum is absolutely continuous. This result solves the Birman problem.

DOI 10.1134/S1061920824030117