Lower bound of Schr{\"o}dinger operators on Riemannian manifolds

TL;DR

This paper establishes lower bounds for Schr{"o}dinger operators on weighted Riemannian manifolds by linking geometric inequalities with spectral estimates, including conditions for positivity and Hardy inequalities.

Contribution

It introduces new lower bounds for Schr{"o}dinger operators on weighted manifolds using Faber-Krahn and Morrey norms, connecting geometry with spectral theory.

Findings

Weighted manifolds satisfying Faber-Krahn inequality admit Fefferman-Phong inequality.

Conditions for L2 Hardy inequality to hold are derived.

Estimates on the bottom of the spectrum of Schr{"o}dinger operators are provided.

Abstract

We show that a weighted manifold which admits a relative Faber Krahn inequality admits the Fefferman Phong inequality V $\psi$, $\psi$ $\le$ CV $\psi$ 2 , with the constant depending on a Morrey norm of V , and we deduce from it a condition for a L 2 Hardy inequality to holds, as well as conditions for Schr{\"o}dinger operators to be positive. We also obtain an estimate on the bottom of the spectrum for Schr{\"o}dinger operators.