A multi-point maximum principle to prove global Harnack inequalities for Schrödinger operators
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
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2025
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| _version_ | 1866914029748879360 |
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| author | Andrews, Ben Hauer, Daniel Slegers, Jessica |
| author_facet | Andrews, Ben Hauer, Daniel Slegers, Jessica |
| contents | In this article, we introduce a new methodology to prove global parabolic Harnack inequalities on Riemannian manifolds. We focus on presenting a new proof of the global pointwise Harnack inequality satisfied by positive solutions of the linear Schrödinger equation on a Riemannian manifold $M$ with nonnegative Ricci curvature, where the potential term $V$ is bounded from below. Our approach is based on a multi-point maximum principle argument. Standard proofs of this result (see, for instance, Li-Yau [Acta Math, 1986]) rely on first establishing a gradient estimate. This requires the solution to be at least $C^4$ on $M$. We instead prove the Harnack inequality directly, which has the advantage of avoiding higher-order derivatives of the solution in the proof, enabling us to assume it is only $C^2$ on $M$. In the particular case that $V$ is the quadratic potential $V(x)=|x|^2$ and $M$ is the Euclidean space $\mathbb{R}^d$, we prove a new Harnack inequality with sharper constants. Finally, we treat positive solutions of the Schrödinger equation with a gradient drift term, including applications to the Ornstein-Uhlenbeck operator $Δ- x\cdot \nabla$ with quadratic potential in $\mathbb{R}^d$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_07575 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A multi-point maximum principle to prove global Harnack inequalities for Schrödinger operators Andrews, Ben Hauer, Daniel Slegers, Jessica Analysis of PDEs In this article, we introduce a new methodology to prove global parabolic Harnack inequalities on Riemannian manifolds. We focus on presenting a new proof of the global pointwise Harnack inequality satisfied by positive solutions of the linear Schrödinger equation on a Riemannian manifold $M$ with nonnegative Ricci curvature, where the potential term $V$ is bounded from below. Our approach is based on a multi-point maximum principle argument. Standard proofs of this result (see, for instance, Li-Yau [Acta Math, 1986]) rely on first establishing a gradient estimate. This requires the solution to be at least $C^4$ on $M$. We instead prove the Harnack inequality directly, which has the advantage of avoiding higher-order derivatives of the solution in the proof, enabling us to assume it is only $C^2$ on $M$. In the particular case that $V$ is the quadratic potential $V(x)=|x|^2$ and $M$ is the Euclidean space $\mathbb{R}^d$, we prove a new Harnack inequality with sharper constants. Finally, we treat positive solutions of the Schrödinger equation with a gradient drift term, including applications to the Ornstein-Uhlenbeck operator $Δ- x\cdot \nabla$ with quadratic potential in $\mathbb{R}^d$. |
| title | A multi-point maximum principle to prove global Harnack inequalities for Schrödinger operators |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2509.07575 |