Eigenvalue Estimates for Schrödinger Operators on Ricci Shrinkers
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866910247536295936 |
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| author | Cheng, Xu Conrado, Franciele Pinheiro, Neilha Zhou, Detang |
| author_facet | Cheng, Xu Conrado, Franciele Pinheiro, Neilha Zhou, Detang |
| contents | Let $(M, g, f, τ)$ be a complete Ricci shrinker satisfying $\textrm{Ric}+\nabla^2f=\frac{g}{2τ}$ 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 $-Δ+\frac{R}{4}+V$, 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 $μ$-functional. We also study the drifted Schrödinger operator $-Δ_f+V$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_23199 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Eigenvalue Estimates for Schrödinger Operators on Ricci Shrinkers Cheng, Xu Conrado, Franciele Pinheiro, Neilha Zhou, Detang Differential Geometry 53C24, 35P15 Let $(M, g, f, τ)$ be a complete Ricci shrinker satisfying $\textrm{Ric}+\nabla^2f=\frac{g}{2τ}$ 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 $-Δ+\frac{R}{4}+V$, 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 $μ$-functional. We also study the drifted Schrödinger operator $-Δ_f+V$ 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. |
| title | Eigenvalue Estimates for Schrödinger Operators on Ricci Shrinkers |
| topic | Differential Geometry 53C24, 35P15 |
| url | https://arxiv.org/abs/2605.23199 |