On a positivity preservation property for Schr\"odinger operators on Riemannian manifolds

TL;DR

This paper investigates a positivity preservation property for Schrödinger operators with singular potentials on complete Riemannian manifolds, and explores its implications for the self-adjointness of these operators.

Contribution

It introduces a positivity preservation property for Schrödinger operators with singular potentials on Riemannian manifolds and applies it to establish self-adjointness criteria.

Findings

Positivity preservation holds for Schrödinger operators with singular potentials.

The property is used to determine self-adjointness of the maximal realization.

Results are applicable to manifolds with non-negative Ricci curvature.

Abstract

We study a positivity preservation property for Schr\"odinger operators with singular potential on geodesically complete Riemannian manifolds with non-negative Ricci curvature. We apply this property to the question of self-adjointness of the maximal realization of the corresponding operator.