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

2010/01/11 by Helffer, Bernard, Kordyukov, Yuri A.
#Analysis of PDEs (math.AP) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1001.1400

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 b₀ 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.

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