Spectral Properties of Image Measures Under the Infinite Conflict Interaction

We introduce the conflict interaction with two positions between a couple of image probability measures and consider the associated dynamical system. We prove the existence of invariant limiting measures and find the criteria for these measures to be a pure point, absolutely continuous, or singular cotinuous as well as to have any topological type and arbitary Hausdorff dimension.     

Schrödinger Operator Perturbed by Operators Related to Null Sets

We discuss the Schrödinger operator with positive singular perturbations given by operators which act in the space constructed by a positive measure supported by a null set. We construct examples when perturbations are given by the one-dimensional Laplacian on a segment.     

Dynamics of conflict interaction between systems with internal structure

We investigate a dynamical system of conflict between two systems each of which, in turn, has an internal conflict. The external conflict and the internal one have different natures. The external conflict is described by an alternative interaction between nonannihilating adversaries. The internal conflict is similar to a conflict between interrelated populations of different biological nature (“predator-prey” model). We construct a computer model of this system and describe a typical behavior, illustrated by diagrams, which can be interpreted, in particular, as a migration of labor and investments between countries. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 4, pp. 435–450, October–December, 2006.     

Decompositions of Singular Continuous Spectra of \mathcal{H}_{ - 2} -class Rank One Perturbations

The decomposition theory for the singular continuous spectrum of rank one singular perturbations is studied. A generalization of the well-known Aronszajn-Donoghue theory to the case of decompositions with respect to α-dimensional Hausdorff measures is given and a characterization of the supports of the α-singular, α-absolutely continuous, and strongly α-continuous parts of the spectral measure of - class rank one singular perturbations is given in terms of the limiting behaviour of the regularized Borel transform.     

Regularized approximations of singular perturbations from the h -2-class

For a sequence of singular perturbations belonging to the H _-1-class and converging to a given singular perturbation from the H _-2-class, we find a method of additive regularization that guarantees the strong resolvent convergence of perturbed operators.     

Additive regularization of singular bilinear forms

Conditions on the additive component of a singular bilinear form are found under which the sum is a regular form. As an application the question of regularization of -shaped forms is considered. The case of tensor products of spaces and forms is considered.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 9, pp. 1199–1204, September, 1990.