Semiclassical asymptotic spectrum of a Hartree-type operator near the upper boundary of spectral clusters

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

“…Moreover, a new integral representation was used to determine the asymptotic solution. Comparing the series of asymptotic eigenvalues obtained in [21]- [23] with eigenvalues (3), we see that they do not contain logarithmic corrections and the spectrum splitting in them occurs in the next approximation.…”

“…Eigenvalue problems similar to (1) for a perturbed two-dimensional resonant oscillator with the perturbing potential given by an integral Hartree-type nonlinearity with a smooth self-action potential were previously considered in [21]- [23], where the asymptotic eigenvalues and asymptotic eigenfunctions near the upper boundaries of a spectral cluster were obtained for a polynomial self-action potential.…”

Semiclassical Asymptotics of the Spectrum of a Two-Dimensional Hartree Type Operator Near Boundaries of Spectral Clusters

“…This paper continues the series of papers devoted to the study of the spectra of Hartree type operators near boundaries of spectral clusters. Earlier, the case of a smooth self-action potential was considered in [4,5], where the potential was given by a second degree polynomial in the squared distance. However an important role in applications is played by Hartree type operators with singular self-action potentials.…”