Eigenvalue estimates for a three-dimensional magnetic Schrödinger operator

Abstract

We consider a magnetic Schrödinger operator H^h=(−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 H^h 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.