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

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

2010-01-11

arxiv.org/abs/1001.1400v1

Mathematics

Spectral Theory

23 pages

Scientific paper

We consider a magnetic Schr\"odinger 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.

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