Semiclassical asymptotics and gaps in the spectra of periodic   Schr\"odinger operators with magnetic wells
  We show that, under some very weak assumption of effective variation for the magnetic field, a periodic Schr\"odinger 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. 
Bernard Helffer, Yuri A. Kordyukov