The periodic magnetic Schr\"odinger operators: spectral gaps and tunneling effect

TL;DR

This paper investigates the spectral properties of periodic magnetic Schrödinger operators on certain manifolds, demonstrating the existence of many spectral gaps in the semiclassical limit through analysis of tunneling effects.

Contribution

It establishes the existence of arbitrarily many spectral gaps for magnetic Schrödinger operators on specific manifolds, advancing understanding of their spectral structure.

Findings

Existence of arbitrarily many spectral gaps in the spectrum.

Spectral gaps are shown to appear in the semiclassical limit.

Analysis of tunneling effects underpins the spectral gap results.

Abstract

A periodic Schr\"odinger operator on a noncompact Riemannian manifold $M$ such that $H^1(M, \mathbb R)=0$ endowed with a properly discontinuous cocompact isometric action of a discrete group is considered. Under some additional conditions on the magnetic field existence of an arbitrary large number of gaps in the spectrum of such an operator in the semiclassical limit is established. The proofs are based on the study of the tunneling effect in the corresponding quantum system.