Asymptotic solutions of Hartree equations concentrated near low-dimensional submanifolds. II. Localization in planar discs

We consider the eigenvalue problem for the three-dimensional Hartree equation in an external field and construct asymptotic (quasi-classical) solutions concentrated near two-dimensional planar discs. The rate of decrease of these solutions along the normal to the disc is determined by the Bogolyubov polaron, and near the edge of the disc it is defined by the Airy analogue of the polaron. To find the related series of eigenvalues, an analogue of the Bohr–Sommerfeld quantization rule is found from which is derived a simpler algebraic equation determining the main terms in the asymptotics of the eigenvalues.