Semiclassical spectral asymptotics for a two-dimensional magnetic Schrödinger operator: The case of discrete wells

Abstract

We consider a magnetic Schrödinger operator $H^h$, depending on the semiclassical parameter $h>0$, on a two-dimensional Riemannian manifold. We assume that there is no electric field. We suppose that the minimal value $b_0$ of the 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 a complete asymptotic expansion for the low-lying eigenvalues of the operator $H^h$ in the semiclassical limit. We also apply these results to prove the existence of an arbitrary large number of spectral gaps in the semiclassical limit in the corresponding periodic setting.

23 pages