Affiliations: Département de Mathématiques, Université Paris-Sud et CNRS, Orsay Cédex, France. E-mail: [email protected] | Institute of Mathematics, Russian Academy of Sciences, Ufa, Russia. E-mail: [email protected]
Note: [] Corresponding author: Yuri A. Kordyukov, Institute of Mathematics, Russian Academy of Sciences, 112 Chernyshevsky str., 450008 Ufa, Russia. Tel.: +7 347 273 33 42; Fax: +7 347 272 59 36; E-mail: [email protected]
Abstract: We consider a magnetic Schrödinger operator Hh=(−ih∇−A¯)2 with the Dirichlet boundary conditions in a domain Ω⊂R3, where h>0 is a small parameter. We suppose that the minimal value b0 of the module |B¯| of the vector magnetic field B¯ is strictly positive, and there exists a unique minimum point of |B¯|, which is non-degenerate. The main result of the paper is upper estimates for the low-lying eigenvalues of the operator Hh in the semiclassical limit. We also prove the existence of an arbitrary large number of spectral gaps in the semiclassical limit in the corresponding periodic setting.
Keywords: magnetic Schrödinger operator, eigenvalue asymptotics, magnetic wells, semiclassical limit, spectral gaps
