Asymptotics of the Hartree-type operator spectrum near the lower boundaries of spectral clusters

Abstract

We consider the eigenvalue problem for a perturbed two-dimensional resonance oscillator. The excitation potential is given by a Hartree-type nonlinearity with a smooth self-action potential. We use asymptotic formulas for the quantum averages to obtain asymptotic eigenvalues and asymptotic eigenfunctions near the lower boundaries of spectral clusters which are formed near the energy levels of the unperturbed operator.