Eigenvalue estimates for a three-dimensional magnetic Schrödinger operator

Details Eigenvalue estimates for a three-dimensional magnetic Schrödinger operator Eigenvalue estimates for a three-dimensional magnetic Schrödinger operator

2012-03-19

arxiv.org/abs/1203.4021v1

Mathematics

Spectral Theory

20 pages

Scientific paper

We consider a magnetic Schr\"odinger operator $H^h=(-ih\nabla-\vec{A})^2$ with the Dirichlet boundary conditions in an open set $\Omega \subset {\mathbb R}^3$, where $h>0$ is a small parameter. We suppose that the minimal value $b_0$ of the module $|\vec{B}|$ of the vector magnetic field $\vec{B}$ is strictly positive, and there exists a unique minimum point of $|\vec{B}|$, 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.

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