Asymptotics of the Spectrum of a Hartree Type Operator with Self-Consistent Potential Including the Macdonald Function

We study the eigenvalue problem for a perturbed two-dimensional oscillator. The role of a perturbation is played by a Hartree type integral nonlinearity, where the self-consistent potential contains the Macdonald function and depends on the distance between points. We find asymptotic eigenvalues and asymptotic eigenfunctions near the upper boundaries of spectral clusters. We construct an asymptotic expansion near a circle, where the solution is localized.