Spectral Gaps for Periodic Schrödinger Operators with Hypersurface Magnetic Wells

We consider a periodic magnetic Schrödinger operator on a noncompact Riemannian manifold M such that H(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 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. Introduction Let M be a noncompact oriented manifold of dimension n ≥ 2 equipped with a properly discontinuous action of a finitely generated, discrete group Γ such that M/Γ is compact. Suppose that H1(M,R) = 0, i.e. any closed 1-form on M is exact. Let g be a Γ-invariant Riemannian metric and B a real-valued Γ-invariant closed 2-form on M . Assume that B is exact and choose a real-valued 1-form A on M such that dA = B. Thus, one has a natural mapping u 7→ ih du+ Au from C∞ c (M) to the space Ω 1 c(M) of smooth, compactly supported oneforms on M . The Riemannian metric allows to define scalar products in these spaces and consider the adjoint operator (ih d + A) : Ωc(M) → C c (M). A Schrödinger operator with magnetic potential A is defined by the formula H = (ih d + A)(ih d+ A). Here h > 0 is a semiclassical parameter, which is assumed to be small.