The spectrum of Schrödinger operators with positive potentials in Riemannian manifolds

Let M be a noncompact complete Riemannian manifold. We consider the Schrödinger operator -Δ + V acting on L²(M), where V is a non-negative, locally integrable function on M. We obtain some simple conditions which imply that inf Spec(-Δ + V), the bottom of the spectrum of -Δ + V, is strictly positive. We also establish upper and lower bounds for the counting function N(λ).