Про резольвенту диференціально-граничного оператора типу Штурма-Ліувілля з багатоточково-інтегральними крайовими умовами

Abstract

Застосовуючи методи функціонального аналізу й теорії диференціальних рівнянь, встановлено умову, яка гарантує приналежність даного комплексного числа до резольвентної множини диференціально-граничного оператора типу Штурма-Ліувілля з багатоточково-інтегральними крайовими умовами та побудовано його резольвенту (іншими словами, знайдено умови, достатні для коректного розв’язання відповідного операторного рівняння та знайдено його розв’язок). Описано множину власних значень досліджуваного оператора.

The problem of construction a resolvent in the theory of perturbation is of special importance. In fact, when the relation between the resolvents of perturbed operator and non-perturbed one, as well as the spectral properties of the first are known, one can obtain some information about the properties of the batter (proper value asymptotics, completeness of the proper values system, spectrum nature, etc). In the case of additive perturbations, this relation is expressed by the Weinstein–Aronszajn type formula, whereas the Krein formula are used in the case, when both the perturbed and the non-perturbed operators are self-adjoint extensions of this symmetric operator. However, these results were obtained under assumption that both the perturbed and the non-perturbed operators have the same domain. Systematic studying of differential-boundary operators with non-classical boundary conditions gave rise to new theoretical operator models in the theory of perturbations, that change not only the operator action law, but also its domain. One of such models is the theory of related operators introduced by V.E. Lyantse who established a connection between resolvents of the related operators. The objective of this paper is to apply the abstract results previously obtained by the author to the case, when the non-perturbed operator is a differential operator acting in the Hilbert space of infinite- dimensional vector-valued functions. Using the methods of functional analysis and differential equations, the condition ensuring that a given complex number belongs to the resolvent set of the differential-boundary Sturm– Liouville type operator with multi-points integral boundary conditions have been established and its resolvent has been built. That is, conditions efficient for the corresponding operator equation solution have been established and the solution has been found. The set of proper values of the operator has been described. The obtained results are theoretical, but they can be applied while solving the operator equations and studying the spectral characteristics of the operators under consideration.