Myers' type theorems and some related oscillation results

TL;DR

This paper extends Myers' type theorems to second order differential equations with negative curvature, providing new compactness and spectral results for solutions and Schrödinger operators on manifolds.

Contribution

It introduces generalized Myers' theorems applicable to equations with negative curvature and explores spectral properties of Schrödinger operators on complete manifolds.

Findings

Existence and localization of zeros lead to compactness results.

Generalized Myers' theorems hold with negative curvature.

Spectral properties of Schrödinger operators are characterized.

Abstract

In this paper we study the behavior of solutions of a second order differential equation. The existence of a zero and its localization allow us to get some compactness results. In particular we obtain a Myers' type theorem even in the presence of an amount of negative curvature. The technique we use also applies to the study of spectral properties of Schroedinger operators on complete manifolds.