Riesz basis property and exponential stability for one-dimensional thermoelastic system with variable coefficients

In this paper, we study Riesz basis property and stability for a nonuniform thermoelastic system with Dirichlet-Dirichlet boundary condition, where the heat subsystem is considered as a control to the whole coupled system. By means of the matrix operator pencil method, we obtain the asymptotic expressions of the eigenpairs, which are exactly coincident to the constant coefficients case} We then show that there exists a sequence of generalized eigenfunctions of the system, which forms a Riesz basis for the state space and the spectrum determined growth condition is therefore proved. As a result, the exponential stability of the system is concluded.

EIGENFUNCTION EXPANSIONS OF THE MAGNETIC SCHRÖDINGER OPERATOR IN BOUNDED DOMAINS.

In this work, we introduce the magnetic Schrödinger operator corresponding to the generalized Dirichlet problem. We prove its self-adjointness and discreteness of the spectrum in bounded domains in the multidimensional case. We also prove the basis property of its eigenfunctions in the Lebesgue space and in the magnetic Sobolev space. We give a new characteristic of the definition domain of the magnetic Schrödinger operator. We investigate the existence and uniqueness of a solution of the magnetic Schrödinger equation with a spectral parameter. It is proved that if the spectral parameter is different from the eigenvalues, then the first generalized Dirichlet problem has a unique solution. We then find the solvability condition for the generalized Dirichlet problem when the spectral parameter coincides with the eigenvalue of the Schrödinger magnetic operator.

Analytical continuation of two-dimensional wave fields

Wave fields obeying the two-dimensional Helmholtz equation on branched surfaces (Sommerfeld surfaces) are studied. Such surfaces appear naturally as a result of applying the reflection method to diffraction problems with straight scatterers bearing ideal boundary conditions. This is for example the case for the classical canonical problems of diffraction by a half-line or a segment. In the present work, it is shown that such wave fields admit an analytical continuation into the domain of two complex coordinates. The branch sets of such continuation are given and studied in detail. For a generic scattering problem, it is shown that the set of all branches of the multi-valued analytical continuation of the field has a finite basis. Each basis function is expressed explicitly as a Green’s integral along so-called double-eight contours. The finite basis property is important in the context of coordinate equations, introduced and used by the authors previously, as illustrated in this article for the particular case of diffraction by a segment.

To the Problem on Small Oscillations of a System of Two Viscoelastic Fluids Filling Immovable Vessel: Model Problem

In this paper, we study the scalar conjugation problem, which models the problem of small oscillations of two viscoelastic fluids filling a fixed vessel. An initial-boundary value problem is investigated and a theorem on its unique solvability on the positive semiaxis is proven with semigroup theory methods. The spectral problem that arises in this case for normal oscillations of the system is studied by the methods of the spectral theory of operator functions (operator pencils). The resulting operator pencil generalizes both the well-known S. G. Kreyns operator pencil (oscillations of a viscous fluid in an open vessel) and the pencil arising in the problem of small motions of a viscoelastic fluid in a partially filled vessel. An example of a two-dimensional problem allowing separation of variables is considered, all points of the essential spectrum and branches of eigenvalues are found. Based on this two-dimensional problem, a hypothesis on the structure of the essential spectrum in the scalar conjugation problem is formulated and a theorem on the multiple basis property of the system of root elements of the main operator pencil is proved.

Completely 0-simple semigroup with the basis property

A semigroup [Formula: see text] is said to have the Basis Property if for any subsemigroup [Formula: see text] of a semigroup [Formula: see text], any two bases for [Formula: see text] have the same cardinality. The structure of completely [Formula: see text]-simple semigroup with the Basis Property is described. In particular, we proved that each completely [Formula: see text]-simple semigroup [Formula: see text] has the Basis Property if and only if [Formula: see text] satisfies one of the following conditions: (1) [Formula: see text] is produced from a completely simple semigroup with adjoint zero. (2) [Formula: see text] is an isomorphic to Rees’s semigroup [Formula: see text] over a group [Formula: see text] with sandwich matrix [Formula: see text] such that [Formula: see text], [Formula: see text], in addition [Formula: see text] has a zero in every row and column.

On a Problem that Does not Have Basis Property of Root Vectors, Associated with a Perturbed Regular Operator of Multiple Differentiation

A spectral problem for a multiple differentiation operator with integral perturbation of boundary value conditions which are regular but not strongly regular is considered in the paper. The feature of the problem is the absence of the basis property of the system of root vectors. A characteristic determinant of the spectral problem is constructed. It is shown that absence of the basis property of the system of root functions of the problem is unstable with respect to the integral perturbation of the boundary value condition