Eigenvalue estimates for a three-dimensional magnetic Schr\"odinger operator

TL;DR

This paper provides upper bounds for low-lying eigenvalues of a 3D magnetic Schrödinger operator with Dirichlet boundary conditions in the semiclassical limit, and demonstrates the existence of many spectral gaps in a periodic setting.

Contribution

It introduces new upper eigenvalue estimates for a magnetic Schrödinger operator in three dimensions with a non-degenerate magnetic field minimum.

Findings

Upper estimates for low-lying eigenvalues in the semiclassical limit

Existence of arbitrarily many spectral gaps in the periodic case

Results depend on the non-degenerate minimum of the magnetic field module

Abstract

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.