The spectrum of Schrödinger operators with positive potentials in Riemannian manifolds

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