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This paper is devoted to a new statement and the study of direct and inverse spectral problems for small linear oscillations of orthotropic plates that carry concentrated masses at a finite set of points, which, in turn, are connected to a stationary base by elastic springs with known stiffness coefficients.

We study the inverse spectral problem of reconstructing nonsplitting boundary conditions for the Sturm–Liouville operator from a minimal amount of spectral data. Necessary and sufficient conditions are derived for the existence of solutions of the problem under consideration and for these solutions to be isolated or nonisolated. We propose a simple analytical algorithm for reconstructing boundary conditions. (Explicit formulas are provided for the coefficients of the boundary conditions.) It is shown that the condition determining the properties of the problem is the dimension of the linear span of matrices of fundamental solution systems corresponding to the given spectral data.

We establish a relationship between an inverse optimization spectral problem for N-dimensional Schr\"odinger equation $ -\Delta \psi+q\psi=\lambda \psi $ and a solution of the nonlinear boundary value problem $-\Delta u+q_0 u=\lambda u- u^{\gamma-1},~~u>0,~~ u|_{\partial \Omega}=0$. Using this relationship, we find an exact solution for the inverse optimization spectral problem, investigate its stability and obtain new results on the existence and uniqueness of the solution for the nonlinear boundary value problem., Comment: 11 pages

The problem of restoration of the parameters of the linear operator from a finite set of eigenvalues was considered. A new scheme of its solution was proposed, both the necessary and sufficient conditions for existence of solutions were presented, and a procedure of their numerical construction was substantiated. Consideration was given to the applied problems of identification and controllability of the linear dynamic systems.

In this paper, we consider the problem on localization of the spectrum of non-self-adjoint differential operators on unbounded domains with power coefficients. To find the location of spectrum points in the complex plane, we use isospectral deformations of differential operators and the properties of families of closed operators analytic in the Kato sense.

and by L0 the minimal operator (see [1]) generated by the expression (1) in L 2 n[0,∞) . Under certain conditions imposed on the matrix A(x) , the operator L0 has deficiency indices equal to (2n, 2n) and it follows that the spectrum of any self-adjoint extension of the operator is discrete. Let L be a real self-adjoint extension, and let λ0 , λ±1 , . . . , λ±k , . . . be the eigenvalues of the operator L arranged in the increasing order of their absolute values. For λ ≥ 0, set N+(L, λ) = ∑ 0≤λk≤λ 1