Asymptotics of the Hartree operator spectrum near the upper boundaries of spectral clusters: Asymptotic solutions localized near a circle

Abstract

We consider the eigenvalue problem for the Hartree operator with a small parameter multiplying the nonlinearity. We obtain asymptotic eigenvalues and asymptotic eigenfunctions near the upper boundaries of spectral clusters formed near the energy levels of the unperturbed operator. Near the circle where the solution is localized, the leading term of the expansion is a solution of the two-dimensional oscillator problem.