Semiclassical analysis of Schr\"odinger operators with magnetic wells

TL;DR

This paper surveys spectral properties of magnetic Schr"odinger operators in the semiclassical limit, focusing on eigenvalue asymptotics and spectral gaps on closed manifolds and periodic operators.

Contribution

It compiles recent results on eigenvalue asymptotics and spectral gaps for magnetic Schr"odinger operators, emphasizing the authors' own contributions.

Findings

Asymptotic behavior of eigenvalues near the bottom of the spectrum.

Existence of spectral gaps in periodic magnetic Schr"odinger operators.

Results mainly obtained by the authors and collaborators.

Abstract

We give a survey of some results, mainly obtained by the authors and their collaborators, on spectral properties of the magnetic Schr\"odinger operators in the semiclassical limit. We focus our discussion on asymptotic behavior of the individual eigenvalues for operators on closed manifolds and existence of gaps in intervals close to the bottom of the spectrum of periodic operators.