The lowest eigenvalue of Schr\"odinger operators on compact manifolds

TL;DR

This paper investigates the lowest eigenvalue of Schr"odinger operators on compact manifolds, especially in the challenging case where the potential changes sign but has a positive average, revealing nuanced spectral properties.

Contribution

It provides new insights into the spectral behavior of Schr"odinger operators with sign-changing potentials on compact manifolds, a less understood area.

Findings

Characterization of the lowest eigenvalue under sign-changing potentials.

Conditions under which the lowest eigenvalue remains positive.

Implications for quantum mechanics on curved spaces.

Abstract

The lowest eigenvalue of the Schr\"odinger operator $-\Delta+\mathcal{V}$ on a compact Riemannian manifold without boundary is studied. We focus on the particularly subtle case of a sign changing potential with positive average.