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We consider a periodic magnetic Schr\"odinger operator on a noncompactRiemannian manifold $M$ such that $H^1(M, \RR)=0$ endowed with a properlydiscontinuous cocompact isometric action of a discrete group. We assume thatthere is no electric field and that the magnetic field has a periodic set ofcompact magnetic wells. We review a general scheme of a proof of existence ofan arbitrary large number of gaps in the spectrum of such an operator in thesemiclassical limit, which was suggested in our previous paper, and someapplications of this scheme. Then we apply these methods to establish similarresults in the case when the wells have regular hypersurface pieces.

Spectral gaps for periodic Schr\"odinger operators with hypersurface magnetic wells