Asymptotic Behavior of the Eigenvalues of the Laplacian with Two Distant Perturbations on the Plane (the Case of Arbitrary Multiplicity)

Abstract

The Laplacian with a pair of distant perturbations is studied in two-dimensional space. Perturbations are understood as real finite continuous potentials. The discrete spectrum of the perturbed Laplacian is studied when the distance between the potentials increases. The presence of its eigenvalues and eigen-functions that correspond to them is considered for various cases of multiplicities of the limiting eigenvalue. The first case of the considered multiplicity is the double limiting eigenvalue. By this we mean the simple and isolated Laplacian eigenvalue with the first potential, as well as the simple and isolated Laplacian eigenvalue with the second potential. The second case under consideration is the case of arbitrary multiplicity of the limiting eigenvalue. By this we mean the Laplacian eigenvalue with the first potential of arbitrary multiplicity and the Laplacian eigenvalue with the second potential also of arbitrary multiplicity. In both cases under consideration (of multiplicity two and arbitrary), the first terms of formal asymptotic expansions of the eigenvalues and eigenfunctions of the perturbed Laplacian are constructed. A complex exponential-power structure of the constructed asymptotic is demonstrated. Also, in both cases under consideration, symmetry with respect to zero of the first corrections of the asymptotics of the eigenvalues of the perturbed Laplacian is shown