The spectrum of Schrödinger operators and Hodge Laplacians on conformally cusp manifolds

Abstract

We describe the spectrum of the k k -form Laplacian on conformally cusp Riemannian manifolds. The essential spectrum is shown to vanish precisely when the k k and k − 1 k-1 de Rham cohomology groups of the boundary vanish. We give Weyl-type asymptotics for the eigenvalue-counting function in the purely discrete case. In the other case we analyze the essential spectrum via positive commutator methods and establish a limiting absorption principle. This implies the absence of the singular spectrum for a wide class of metrics. We also exhibit a class of potentials V V such that the Schrödinger operator has compact resolvent, although in most directions the potential V V tends to − ∞ -\infty . We correct a statement from the literature regarding the essential spectrum of the Laplacian on forms on hyperbolic manifolds of finite volume, and we propose a conjecture about the existence of such manifolds in dimension 4 whose cusps are rational homology spheres.