Semiclassical spectral asymptotics for a two-dimensional magnetic Schr\"odinger operator: The case of discrete wells

TL;DR

This paper derives detailed asymptotic formulas for low-energy eigenvalues of a 2D magnetic Schrödinger operator on a Riemannian manifold with a unique magnetic field minimum, and demonstrates the existence of many spectral gaps in the periodic case.

Contribution

It provides a complete asymptotic expansion for eigenvalues of the magnetic Schrödinger operator with discrete wells, extending spectral analysis in the semiclassical limit.

Findings

Asymptotic expansion for low-lying eigenvalues

Existence of arbitrarily many spectral gaps in periodic setting

Analysis under non-degenerate magnetic field minimum

Abstract

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.