Дуальна пара власних значень сингулярно несиметрично рангу один збурених операторів

Вдовенко, Тетяна

Дуальна пара власних значень сингулярно несиметрично рангу один збурених операторів

Date

2016

Authors

Вдовенко, Тетяна

Abstract

Розглядається задача на власнi значення сингулярного несамоспряженого збурення рангу один самоспряженого оператора A несиметричним потенцiалом (δ1 6= δ2) у виглядi A˜ = A+αh·, δ1iδ2. Надається конструтивний опис оператора вигляду A˜, що має двi новi точки точкового спектра у випадку слабо сингулярного збурення.

Description

We discuss the eigenvalue problem for a rank one singular non-selfadjoint perturbation of a selfadjoint operator A in the separable Hilbert space, by nonsymmetric potential (δ1 6= δ2) in the form A˜ = A + + αh·, δ1iδ2. We give the constructive description of such sort operator A˜ which possess two new points in the point spectrum in case of weakly singular perturbations. In our investigations we can really observe the pair of symmetric operator with defect indexes (1,1) both and consider only some class of nonsymmetric extensions. There are known old questions: under what conditions the Schr¨odinger operator have a point spectrum immersed in the continuous one, is difficult from a physical point of view. The study of this case is particularly unpromising, because there are good physical reasons to expect that such eigenvalues should not be. However, there are known examples of J. von Neumann (1929) in which described Hamiltonian perturbed by free smooth potentials such that perturbed operator becomes coherent states inside the continuous spectrum. The main considerations of this kind of cases mainly focused on how to avoid appearance of eigenvalues embedded in the continuous spectrum, as this creates difficulties by researches in the scattering theory. But the work S. Albeverio, M. Dudkin, V. Koshmanenko contains the description on an unexpected appearance: rank one singularly perturbed self-adjoint operator possess two new eigenvalues so that one of them is immersed in a continuous spectrum of the unperturbed (given) operator. Since the study of singular perturbation operators extended to perturbations nonsymmetric potentials, you should expect also associated pairs by rank one singularly perturbed In fact, we investigate the inverse eigenvalue problem for perturbations of nonsymmetric potentials. Namely, we present perturbation A˜ which solves the eigenvalue problem for the dual pair λ, μ ∈ C: A˜ϕλ = λϕλ, ˜ Aϕμ = μϕμ, ˜ A∗ψ¯λ = ¯λ ψ¯λ, ˜ A∗ψ¯μ = ¯μψ¯μ, (¯λ − ¯μ)((A − μ)−1ϕλ, ψ¯λ ) = (ϕλ, ψ¯λ ).

Keywords

сингулярнi збурення рангу один, задача на власнi значення, формула М. Крейна, несамоспряжене збурення, аргумент вiдхилення, стаття, rank one singular perturbation, eigenvalue problem, M. Krein’s formula, nonselfadjoint perturbation, deviating argument

Citation

Вдовенко Т. І. Дуальна пара власних значень сингулярно несиметрично рангу один збурених операторів / Вдовенко Т. І. // Наукові записки НаУКМА : Фізико-математичні науки. - 2016. - Т. 178. - С. 3-9.