On Laplacian Discrete Spectrum Behavior with Two Distant Perturbations on the Plane in the case of a double limiting eigenvalue

Abstract

We consider the Laplacian with two distant perturbations on the plane. Perturbations are a real finite continuous potentials. The investigation is aimed at a behavior of the perturbed operator eigenvalues when the distance between the potentials tends to infinity. The study concerns an existence of the perturbed eigenvalues in the case of a double limiting eigenvalue (a double eigenvalue of the Laplacian with the first finite potential).The paper aim is to construct the first terms of the asymptotic expansions of the perturbed eigenvalues and the corresponding eigenfunctions in the case of a double limiting eigenvalue.The technique used to obtain the results is also applicable to the construction of complete asymptotic expansions of perturbed eigenvalues аnd their corresponding eigenfunctions. The finiteness of the distant potentials made it possible to reveal the complex exponential-power structure of the asymptotics obtained.The main study results include  the following:the first terms of the asymptotic expansions of the perturbed eigenvalues and their corresponding eigenfunctions;the first corrections of the asymptotics of the perturbed eigenvalues being equal to zero;exponential-power structure of the asymptotics of perturbed eigenvalues and their corresponding eigenfunctions.