Eigenvalue Estimates for Schr\"odinger Operators on Ricci Shrinkers

Let $(M, g, f, \tau)$ be a complete Ricci shrinker satisfying $\textrm{Ric}+\nabla^2f=\frac{g}{2\tau}$ 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 Schr\"odinger operator $-\Delta+\frac{R}{4}+V$, 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 $\mu$-functional. We also study the drifted Schr\"odinger operator $-\Delta_f+V$ 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.