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.

Keywords:

• Self-consistent field
• spectral cluster
• coherent transformation
• WKB-approximation
• quantum average

AMS Subject Classifications:

• 35C15
• 35C20
• 35Q40
• 33C45

Acknowledgements

The author is grateful to M. V. Karasev for drawing his attention to this problem and also for valuable questions and comments.

Notes

No potential conflict of interest was reported by the author.

Additional information

Funding

The results of the work are obtained in the framework of the state order of the Ministry of Education and Science of the Russian Federation [project number 1.756.2014/K].