# Vol. 22 (2016), no. 1

### The investigation of Bogoliubov functionals by operator methods of moment problem

Yu. M. Berezansky, V. A. Tesko

MFAT **22** (2016), no. 1, 1-47

1-47

The article is devoted to a study of Bogoliubov functionals by using methods of the operator spectral theory being applied to the classical power moment problem. Some results, similar to corresponding ones for the moment problem, where obtained for such functionals. In particular, the following question was studied: under what conditions a sequence of nonlinear functionals is a sequence of Bogoliubov functionals.

### On the finiteness of the discrete spectrum of a 3x3 operator matrix

MFAT **22** (2016), no. 1, 48-61

48-61

An operator matrix $H$ associated with a lattice system describing three particles in interactions, without conservation of the number of particles, is considered. The structure of the essential spectrum of $H$ is described by the spectra of two families of the generalized Friedrichs models. A symmetric version of the Weinberg equation for eigenvectors of $H$ is obtained. The conditions which guarantee the finiteness of the number of discrete eigenvalues located below the bottom of the three-particle branch of the essential spectrum of $H$ is found.

### Joint functional calculus in algebra of polynomial tempered distributions

MFAT **22** (2016), no. 1, 62-73

62-73

In this paper we develop a functional calculus for a countable system of generators of contraction strongly continuous semigroups. As a symbol class of such calculus we use the algebra of polynomial tempered distributions. We prove a differential property of constructed calculus and describe its image with the help of the commutant of polynomial shift semigroup. As an application, we consider a function of countable set of second derivative operators.

### On a generalization of the three spectral inverse problem

O. P. Boyko, O. M. Martynyuk, V. N. Pivovarchik

MFAT **22** (2016), no. 1, 74-80

74-80

We consider a generalization of the three spectral inverse problem, that is, for given spectrum of the Dirichlet-Dirichlet problem (the Sturm-Liouville problem with Dirichlet conditions at both ends) on the whole interval $[0,a]$, parts of spectra of the Dirichlet-Neumann and Dirichlet-Dirichlet problems on $[0,a/2]$ and parts of spectra of the Dirichlet-Newman and Dirichlet-Dirichlet problems on $[a/2,a]$, we find the potential of the Sturm-Liouville equation.

### Dynamical systems of conflict in terms of structural measures

Volodymyr Koshmanenko, Inga Verygina

MFAT **22** (2016), no. 1, 81-93

81-93

We investigate the dynamical systems modeling conflict processes between a pair of opponents. We assume that opponents are given on a common space by distributions (probability measures) having the similar or self-similar structure. Our main result states the existence of the controlled conflict in which one of the opponents occupies almost whole conflicting space. Besides, we compare conflicting effects stipulated by the rough structural approximation under controlled redistributions of starting measures.