Asymptotic Solutions of Two-Dimensional Hartree-Type Equations Localized in the Neighborhood of Line Segments

We consider the eigenvalue problem for the two-dimensional Schrödinger equation containing an integral Hartree-type nonlinearity with an interaction potential having a logarithmic singularity. Global asymptotic solutions localized in the neighborhood of a line segment in the plane are constructed using the matching method for asymptotic expansions. The Bogoliubov and Airy polarons are used as model functions in these solutions. An analogue of the Bohr–Sommerfeld quantization rule is established to find the related series of eigenvalues.