Spectral Gaps for Periodic Schrödinger Operators with Hypersurface Magnetic Wells: Analysis near the Bottom

We consider a periodic magnetic Schrödinger operator H, depending on the semiclassical parameter h > 0, 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 suppose that the magnetic field vanishes regularly on a hypersurface S. First, we prove upper and lower estimates for the bottom λ0(H) of the spectrum of the operator Hin L(M). Then, assuming the existence of non-degenerate miniwells for the reduced spectral problem on S, we prove the existence of an arbitrary large number of spectral gaps for the operator H in the region close to λ0(H), as h → 0. In this case, we also obtain upper estimates for the eigenvalues of the one-well problem. 1. Preliminaries and main results 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 H(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 one-forms 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. Choose local coordinates X = (X1, . . . , Xn) on M . Write the 1-form A in the local coordinates as A = n ∑