Russian Journal of Mathematical Physics最新文献

Recently we have shown that the equivalence classes of metrics on the double of a metric space (X) form an inverse semigroup. Here we define an inverse subsemigroup related to a family of isometric subspaces of (X), which is more computable. As a special case, we study this subsemigroup related to the family of geodesic rays starting from the basepoint, for Euclidean spaces and for trees.

Properties of local solarity and regularity in essentially asymmetric locally uniformly rotund spaces are studied. The results obtained are applied to the study of smooth solutions of the eikonal equation (|nabla u|equiv 1). For this purpose, sets of regular points are investigated. Examples of the influence of caustics on the evolution of elliptical galaxies into spiral ones are given.

We consider a generalization of the mKdV equation which contains dissipation terms similar to those contained in both the Benjamin–Bona–Mahoney equation and the famous Camassa–Holm and Degasperis–Procesi equations. Our objective is the construction of classical (solitons) and non-classical (peakons and cuspons) solitary wave solutions of this equation.

A semiclassical asymptotics of eigenfunctions and eigenvalues is constructed for the one-dimensional Schrödinger operator in which the potential rapidly changes around a certain point.

Boundary value problems are considered in which the main operator and the operators of boundary conditions include differential and shift operators corresponding to the action of a discrete group. The manifold on which the boundary value problem is considered is not assumed to be group invariant. A definition of trajectory symbols for this class of boundary value problems is given. It is shown that elliptic problems define Fredholm operators in the corresponding Sobolev spaces. An application to problems with extensions and contractions is given.

In the present paper, we develop a general mathematical framework for discrimination between (rgeq2) quantum states by (Ngeq1) sequential receivers for the case in which every receiver obtains a conclusive result. This type of discrimination constitutes an (N)-sequential extension of the minimum-error discrimination by one receiver. The developed general framework, which is valid for a conclusive discrimination between any number (rgeq2) of quantum states, pure or mixed, of an arbitrary dimension and any number (Ngeq1) of sequential receivers, is based on the notion of a quantum state instrument, and this allows us to derive new important general results. In particular, we find a general condition on (rgeq2) quantum states under which, within the strategy in which all types of receivers’ quantum measurements are allowed, the optimal success probability of the (N)-sequential conclusive discrimination between these (rgeq2) states is equal to that of the first receiver for any number (Ngeq2) of further sequential receivers and specify the corresponding optimal protocol. Furthermore, we extend our general framework to include an (N)-sequential conclusive discrimination between (rgeq2) arbitrary quantum states under a noisy communication. As an example, we analyze analytically and numerically a two-sequential conclusive discrimination between two qubit states via depolarizing quantum channels. The derived new general results are important both from the theoretical point of view and for the development of a successful multipartite quantum communication via noisy quantum channels.

We consider linear equations with shifts of the arguments on the rectangular lattice with small step (h) in (mathbb{R}^n) and construct a version of the canonical operator providing semiclassical asymptotics for such equations. Examples include the Feynman checkers model arising in quantum theory and a problem on the wave packet propagation on a homogeneous tree.

The Bochner–Schrödinger operator (H_{p}=frac 1pDelta^{L^potimes E}+V) on tensor powers (L^p) of a Hermitian line bundle (L) twisted by a Hermitian vector bundle (E) on a Riemannian manifold of bounded geometry is studied. For any function (varphiin mathcal S(mathbb R)), we consider the bounded linear operator (varphi(H_p)) in (L^2(X,L^potimes E)) defined by the spectral theorem and describe an asymptotic expansion of its smooth Schwartz kernel in a fixed neighborhood of the diagonal in the semiclassical limit (pto infty). In particular, we prove that the trace of the operator (varphi(H_p)) admits a complete asymptotic expansion in powers of (p^{-1/2}) as (pto infty). We also prove a result on the asymptotic localization of the Schwartz kernel of the spectral projection on the diagonal in the case when the curvature is of full rank.

In this paper, we construct and study a model of phase transition in a system of two phases (liquid and ice) and three media, namely, water, a piece of ice, and a nonmelting solid substrate. Namely, the melting-crystallization process is considered in the problem of water flow along a small ice irregularity (such as a frozen drop) on a flat substrate for large Reynolds numbers. The results of numerical simulation are presented.

We discuss homogenization of a strongly elliptic operator (mathcal A^varepsilon=-operatorname{div}A(x,x/varepsilon_#)nabla) on a bounded (C^{1,1}) domain in (mathbb R^d) with either Dirichlet or Neumann boundary condition. The function (A) is piecewise Lipschitz in the first variable and periodic in the second one, and the function (varepsilon_#) is identically equal to (varepsilon_i(varepsilon)) on each piece (Omega_i), with (varepsilon_i(varepsilon)to0) as (varepsilonto0). For (mu) in a resolvent set, we show that the resolvent ((mathcal A^varepsilon-mu)^{-1}) converges, as (varepsilonto0), in the operator norm on (L_2(Omega)^n) to the resolvent ((mathcal A^0-mu)^{-1}) of the effective operator at the rate (varepsilon_ {vee} ), where (varepsilon_ {vee} ) stands for the largest of (varepsilon_i(varepsilon)). We also obtain an approximation for the resolvent in the operator norm from (L_2(Omega)^n) to (H^1(Omega)^n) with error of order (varepsilon_ {vee} ^{1/2}).