Spectral positivity and Riemannian coverings

TL;DR

This paper investigates the relationship between spectral positivity of Schrödinger operators on Riemannian manifolds and their Riemannian coverings, establishing conditions under which positivity properties are preserved or reflected.

Contribution

It proves that spectral positivity on a base manifold implies positivity on a covering manifold if the fundamental group of the cover is co-amenable in the base's fundamental group.

Findings

Positivity of the operator on the base implies positivity on the cover.

The converse holds if the fundamental group of the cover is co-amenable.

Provides a criterion linking algebraic properties of groups to spectral properties of operators.

Abstract

Let $(M,g)$ be a complete non-compact Riemannian manifold. We consider operators of the form $\Delta_g + V$, where $\Delta_g$ is the non-negative Laplacian associated with the metric $g$, and $V$ a locally integrable function. Let $\rho : (\hat{M},\hat{g}) \to (M,g)$ be a Riemannian covering, with Laplacian $\Delta_{\hat{g}}$ and potential $\hat{V} = V \circ \rho$. If the operator $\Delta + V$ is non-negative on $(M,g)$, then the operator $\Delta_{\hat{g}} + \hat{V}$ is non-negative on $(\hat{M},\hat{g})$. In this note, we show that the converse statement is true provided that $\pi_1(\hat{M})$ is a co-amenable subgroup of $\pi_1(M)$.