Dudkin М.E.

Rank One Strong Singular Perturbation by Nonsymmetric Potential

For a rank one strong singular perturbation of a self-adjoint operator by nonsymmetric potential, we present a construction and investigated the corresponding eigenvalue problem.

Direct Spectral Problems for the Block Jacobi Type Bounded Symmetric Matrices Related to the two dimensional real moment problem

The generalization of the classical moment problem and the spectral theory of self-adjoint Jacobi block matrix are well-known in one-dimensional case and it generalized on the two-dimensional case. Finite and infinite moment problem is solved using Yu.M. Berezansky generalized eigenfunction expansion method for respectively finite and infinite family of commuting self-adjoint operators. In the classical case one orthogonalize a family of polynomials , with respect to a measure on the real axis and shift operator on takes the form of ordinary Jacobi matrix.

Spectral Properties of Singularly Perturbed qs-Normal Operator

Using described singularly perturbed rank one qs-normal operator, we study their spectral properties. We construct the singularly perturbed qs-normal operator with the prescribed set of eigenvectors and eigenvalues. When constructing this operator, we use the previously proven theorem on the structure of singularly perturbed self-adjoint operators with prescribed set of eigenvalues and corresponding eigenvectors. In such case the eigenvalues are located on the real axis. Its construction was a step-by-step process.

The Inverse Spectral Problem for the Block Jacobi-Type Matrices in the Complex Moments Problem in the Exponential Form

The article proposes the analog of Jacobi matrices related to the complex moments problem in the case of exponential form as well as to the system of orthonormal polynomials relative to some measure with the compact support on the complex plane. We obtain two matrices having block tridiagonal structure and acting in the space of type as a self-adjoint and unitary commuting operators. The previous research is incomplete as far as this research is concerned.