Vol 75, No 3 (2020)

Generalizations and refinements are given for results of Kozlov and Treschev on non-uniform averagings in the ergodic theorem in the case of operator semigroups on spaces of integrable functions and semigroups of measure-preserving transformations. Conditions on the averaging measures are studied under which the averages converge for broad classes of integrable functions.Bibliography: 96 items.

Linear systems of differential equations in a Hilbert space are considered that admit a positive-definite quadratic form as a first integral. The following three closely related questions are the focus of interest in this paper: the existence of other quadratic integrals, the Hamiltonian property of a linear system, and the complete integrability of such a system. For non-degenerate linear systems in a finite-dimensional space essentially exhaustive answers to all these questions are known. Results of a general nature are applied to linear evolution equations of mathematical physics: the wave equation, the Liouville equation, and the Maxwell and Schrödinger equations.Bibliography: 60 titles.

A waveguide occupies a domain in an $(n+1)$-dimensional Euclidean space which has several cylindrical outlets to infinity. Three classes of waveguides are considered: those of quantum theory, of electromagnetic theory, and of elasticity theory, described respectively by the Helmholtz operator, the Maxwell system, and the system of equations for an elastic medium. It is assumed that the coefficients of all problems stabilize exponentially at infinity, to functions that are independent of the axial variable in the corresponding cylindrical outlet. Each row of the scattering matrix is given approximately by minimizing a quadratic functional. This functional is constructed by use of an elliptic boundary value problem in a bounded domain obtained by cutting the cylindrical outlets of the waveguide at some distance $R$. The existence and uniqueness of a solution is proved for each of the three types of waveguides. The minimizers converge exponentially fast as functions of $R$, as $R\to\infty$, to rows of the scattering matrix.Bibliography: 47 titles.