Spectral gaps for periodic Schr\"odinger operators with hypersurface magnetic wells: Analysis near the bottom

TL;DR

This paper analyzes the spectral properties of a periodic magnetic Schr"odinger operator on a noncompact manifold, establishing estimates for the spectrum's bottom and demonstrating the existence of multiple spectral gaps near the lowest eigenvalue as the semiclassical parameter approaches zero.

Contribution

It provides new estimates for the spectral bottom and proves the existence of infinitely many spectral gaps under certain conditions on the magnetic wells and the hypersurface where the magnetic field vanishes.

Findings

Upper and lower bounds for the spectrum's bottom $ le$ of the operator.

Existence of arbitrarily many spectral gaps near the bottom of the spectrum as $h o 0$.

Upper estimates for eigenvalues of the one-well reduced spectral problem.

Abstract

We consider a periodic magnetic Schr\"odinger operator $H^h$, depending on the semiclassical parameter $h>0$, 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. We assume that there is no electric field and that the magnetic field has a periodic set of compact magnetic wells. We suppose that the magnetic field vanishes regularly on a hypersurface $S$. First, we prove upper and lower estimates for the bottom $\lambda_0(H^h)$ of the spectrum of the operator $H^h$in $L^2(M)$. Then, assuming the existence of non-degenerate miniwells for the reduced spectral problem on $S$, we prove the existence of an arbitrary large number of spectral gaps for the operator $H^h$ in the region close to $\lambda_0(H^h)$, as $h\to 0$. In this case, we also obtain upper estimates for the eigenvalues of the one-well problem.