Spectral gaps for periodic Schrödinger operators with hypersurface magnetic wells: Analysis near the bottom

We consider a periodic magnetic Schrödinger operator Hh, depending on the semiclassical parameter h > 0, on a noncompact Riemannian manifold M such that H1(M,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 λ0(Hh) of the spectrum of the operator Hh in L2(M). Then, assuming the existence of non-degenerate miniwells for the reduced spectral problem on S, we prove the existence of an arbitrarily large number of spectral gaps for the operator Hh in the region close to λ0(Hh), as h→0. In this case, we also obtain upper estimates for the eigenvalues of the one-well problem. © 2009 Elsevier Inc. All rights reserved.