Asymptotics of The Spectrum of a Two-Dimensional Hartree-Type Operator with a Coulomb Self-Action Potential Near the Lower Boundaries of Spectral Clusters

Abstract

We consider the eigenvalue problem for a perturbed two-dimensional oscillator where the perturbation is an integral Hartree-type nonlinearity with a Coulomb self-action potential. We obtain asymptotic eigenvalues and asymptotic eigenfunctions near the lower boundaries of spectral clusters formed in a neighborhood of the eigenvalues of the unperturbed operator and construct an asymptotic expansion near a circle where the solution is localized.