Pereskokov, A. V. Asymptotic solutions of two-dimensional Hartree-type equations localized in the neighborhood of line segments. (English. Russian original) Zbl 1044.35042 Theor. Math. Phys. 131, No. 3, 775-790 (2002); translation from Teor. Mat. Fiz. 131, No. 3, 389-406 (2002). Summary: 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 Bogolyubov 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. Cited in 5 Documents MSC: 35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs 81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory Keywords:semiclassical asymptotic solutions; quantization rule; Hartree equation; Airy polaron PDFBibTeX XMLCite \textit{A. V. Pereskokov}, Theor. Math. Phys. 131, No. 3, 775--790 (2002; Zbl 1044.35042); translation from Teor. Mat. Fiz. 131, No. 3, 389--406 (2002) Full Text: DOI