Semiclassical asymptotics and gaps in the spectra of periodic Schrodinger operators with magnetic wells

We show that, under some very weak assumption of effective variation for the magnetic field, a periodic Schrodinger operator with magnetic wells on a noncompact Riemannian manifold M such that H-1(M, R) = 0, equipped with a properly disconnected, cocompact action of a finitely generated, discrete group of isometries, has an arbitrarily large number of spectral gaps in the semi-classical limit.