Spectral gaps for periodic Schr\"odinger operators with hypersurface magnetic wells

TL;DR

This paper investigates the spectral gaps of periodic magnetic Schrödinger operators on noncompact manifolds with magnetic wells, extending previous methods to cases with hypersurface-shaped wells in the semiclassical limit.

Contribution

It generalizes existing spectral gap results to operators with hypersurface magnetic wells, providing a framework for analyzing spectral properties in this setting.

Findings

Existence of arbitrarily large spectral gaps in the semiclassical limit.

Extension of spectral gap results to hypersurface magnetic wells.

Application of the general scheme to new geometric configurations.

Abstract

We consider a periodic magnetic Schr\"odinger operator on a noncompact Riemannian manifold $M$ such that $H^1(M, \RR)=0$ endowed with a properly discontinuous cocompact isometric action of a discrete group. We assume that there is no electric field and that the magnetic field has a periodic set of compact magnetic wells. We review a general scheme of a proof of existence of an arbitrary large number of gaps in the spectrum of such an operator in the semiclassical limit, which was suggested in our previous paper, and some applications of this scheme. Then we apply these methods to establish similar results in the case when the wells have regular hypersurface pieces.