The spectrum of Schrödinger operators with positive potentials in Riemannian manifolds

Abstract

Let M M be a noncompact complete Riemannian manifold. We consider the Schrödinger operator − Δ + V -\Delta +V acting on L 2 ( M ) L^{2}(M) , where V V is a nonnegative, locally integrable function on M M . We obtain some simple conditions which imply that inf Spec ( − Δ + V ) \inf \text {Spec} (-\Delta +V) , the bottom of the spectrum of − Δ + V -\Delta +V , is strictly positive. We also establish upper and lower bounds for the counting function N ( λ ) N(\lambda ) .