Lower bound of Schrödinger operators on Riemannian manifolds

Abstract

We show that a complete weighted manifold which satisfies to a relative Faber–Krahn inequality admits a trace inequality for the measure with density $V$ , with the constant depending on a Morrey norm of $V$ . From this, we obtain estimates on the lower bound of the spectrum of the Schrödinger operators with potential $V$ and positivity conditions for such operators. It also yields a $L^2$ Hardy inequality.