Spectral gaps for periodic Schrödinger operators with hypersurface magnetic wells

Details Spectral gaps for periodic Schrödinger operators with hypersurface magnetic wells Spectral gaps for periodic Schrödinger operators with hypersurface magnetic wells

2008-01-29

arxiv.org/abs/0801.4460v1

Mathematics

Spectral Theory

16 pages

Scientific paper

We consider a periodic magnetic Schr\"odinger operator on a noncompact Riemannian manifold $M$ such that $H^1(M, \RR)=0$ 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 review a general scheme of a proof of existence of an arbitrary large number of gaps in the spectrum of such an operator in the semiclassical limit, which was suggested in our previous paper, and some applications of this scheme. Then we apply these methods to establish similar results in the case when the wells have regular hypersurface pieces.

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