2018
Том 70
№ 1

# Dudkin M. Ye.

Articles: 4
Article (Ukrainian)

### Jacobi-type block matrices corresponding to the two-dimensional moment problem: polynomials of the second kind and Weyl function

Ukr. Mat. Zh. - 2016. - 68, № 4. - pp. 495-505

We continue our investigations of Jacobi-type symmetric matrices corresponding to the two-dimensional real power moment problem. We introduce polynomials of second kind and the corresponding analog of the Weyl function.

Brief Communications (Ukrainian)

### Point spectrum of the schrödinger operator with point interactions at the vertices of regular N-gons

Ukr. Mat. Zh. - 2004. - 56, № 8. - pp. 1128–1134

We present a complete description of the point spectrum of the Laplace operator perturbed by point potentials concentrated at the vertices of regular polygons. We prove a criterion for the absence of points of the point spectrum of a singular perturbed positive self-adjoint operator with the property of cyclicity of defect vectors.

Brief Communications (Ukrainian)

### On the Point Spectrum of Self-Adjoint Operators That Appears under Singular Perturbations of Finite Rank

Ukr. Mat. Zh. - 2003. - 55, № 9. - pp. 1269-1276

We discuss purely singular finite-rank perturbations of a self-adjoint operator A in a Hilbert space ℋ. The perturbed operators $\tilde A$ are defined by the Krein resolvent formula $(\tilde A - z)^{ - 1} = (A - z)^{ - 1} + B_z$ , Im z ≠ 0, where B z are finite-rank operators such that dom B z ∩ dom A = |0}. For an arbitrary system of orthonormal vectors $\{ \psi _i \} _{i = 1}^{n < \infty }$ satisfying the condition span |ψ i } ∩ dom A = |0} and an arbitrary collection of real numbers ${\lambda}_i \in {\mathbb{R}}^1$ , we construct an operator $\tilde A$ that solves the eigenvalue problem $\tilde A\psi _i = {\lambda}_i {\psi}_i , i = 1, \ldots ,n$ . We prove the uniqueness of $\tilde A$ under the condition that rank B z = n.

Brief Communications (Ukrainian)

### Analog of the Krein Formula for Resolvents of Normal Extensions of a Prenormal Operator

Ukr. Mat. Zh. - 2002. - 54, № 4. - pp. 555-562

We prove a formula that relates resolvents of normal operators that are extensions of a certain prenormal operator. This formula is an analog of the Krein formula for resolvents of self-adjoint extensions of a symmetric operator. We describe properties of the defect subspaces of a prenormal operator.