Spectral properties of Schr\"odinger operators on compact manifolds: rigidity, flows, interpolation and spectral estimates

TL;DR

This paper establishes optimal spectral estimates for Schr"odinger operators on compact manifolds using interpolation inequalities and rigidity results, advancing understanding of spectral properties in geometric analysis.

Contribution

It introduces new spectral estimates for Schr"odinger operators on manifolds, combining interpolation inequalities with recent rigidity results for nonlinear elliptic equations.

Findings

Derived optimal spectral bounds for Schr"odinger operators.

Connected spectral estimates with geometric rigidity results.

Enhanced understanding of spectral behavior on compact manifolds.

Abstract

This note is devoted to optimal spectral estimates for Schr\"odinger operators on compact connected Riemannian manifolds without boundary. These estimates are based on the use of appropriate interpolation inequalities and on some recent rigidity results for nonlinear elliptic equations on those manifolds.