Eigenvalue Estimates for Schrödinger Operators on Ricci Shrinkers

Abstract

Let $(M,g,f,\ddot{I}„)$ be a complete Ricci shrinker satisfying $\text{Ric}+{\nabla }^{2}f=\frac{g}{2\ddot{I}„}$ and let $R$ denote its scalar curvature. For a confined function $V$ on $M$, we obtain a lower bound for the lowest eigenvalue of the Schrödinger operator $-\hat{I}”+\frac{R}{4}+V$, expressed in terms of an integral quantity involving $V$ and the shrinker entropy, and the equality case is characterized by the potential functions. We further generalize this estimate to complete Riemannian manifolds via Perelman's $\hat{I}¼$-functional. We also study the drifted Schrödinger operator $-\hat{I}{”}_f+V$ on smooth metric measure spaces. In particular, on Ricci shrinkers, we derive a lower bound for its lowest eigenvalue, with equality if and only if $V$ is affine.

18 pages. Comments are welcome