THE SPECTRUM OF k-FORM SCHRÖDINGER LAPLACIANS ON CONFORMALLY CUSP MANIFOLDS

We describe the spectrum of the k-form Laplacian on conformally cusp Riemannian manifolds. The essential spectrum is shown to vanish precisely when the k and 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. We also exhibit a class of potentials V such that the Schrödinger operator has compact resolvent, although V tends to −∞ in most of the infinity.