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

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.