Molecules as metric measure spaces with Kato-bounded Ricci curvature
Set $\Psi :=\log (\tilde{\Psi })$, with $\tilde{\Psi }>0$ the ground state of an arbitrary molecule with $n$ electrons in the infinite mass limit (neglecting spin/statistics). Let $\Sigma \subset \mathbb{R}^{3n}$ be the set of singularities of the underlying Coulomb potential. We show that the me...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Académie des sciences
2020-09-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.76/ |
| _version_ | 1797651557942034432 |
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| author | Güneysu, Batu von Renesse, Max |
| author_facet | Güneysu, Batu von Renesse, Max |
| author_sort | Güneysu, Batu |
| collection | DOAJ |
| description | Set $\Psi :=\log (\tilde{\Psi })$, with $\tilde{\Psi }>0$ the ground state of an arbitrary molecule with $n$ electrons in the infinite mass limit (neglecting spin/statistics). Let $\Sigma \subset \mathbb{R}^{3n}$ be the set of singularities of the underlying Coulomb potential. We show that the metric measure space $\mathscr{M}$ given by $\mathbb{R}^{3n}$ with its Euclidean distance and the measure
\[ \mu (\mathrm{d}x)=e^{-2\Psi (x)}\mathrm{d}x \]
has a Bakry-Emery-Ricci tensor which is absolutely bounded by the function $x\mapsto |x-\Sigma |^{-1}$, which we show to be an element of the Kato class induced by $\mathscr{M}$. In addition, it is shown that $\mathscr{M}$ is stochastically complete, that is, the Brownian motion which is induced by a molecule is nonexplosive. Our proofs reveal a fundamental connection between the above geometric/probabilistic properties and recently obtained derivative estimates for $\tilde{\Psi }$ by Fournais/Sørensen, as well as Aizenman/Simon’s Harnack inequality for Schrödinger operators. Moreover, our results suggest to study general metric measure spaces having a Ricci curvature which is synthetically bounded from below/above by a function in the underlying Kato class. |
| first_indexed | 2024-03-11T16:17:32Z |
| format | Article |
| id | doaj.art-aae862fef23f4228b7482d7b3f148b26 |
| institution | Directory Open Access Journal |
| issn | 1778-3569 |
| language | English |
| last_indexed | 2024-03-11T16:17:32Z |
| publishDate | 2020-09-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj.art-aae862fef23f4228b7482d7b3f148b262023-10-24T14:18:59ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692020-09-01358559560210.5802/crmath.7610.5802/crmath.76Molecules as metric measure spaces with Kato-bounded Ricci curvatureGüneysu, Batu0von Renesse, Max1Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, GermanyFakultät für Mathematik und Informatik, Universität Leipzig, Ritterstraße 26, 04109 Leipzig, GermanySet $\Psi :=\log (\tilde{\Psi })$, with $\tilde{\Psi }>0$ the ground state of an arbitrary molecule with $n$ electrons in the infinite mass limit (neglecting spin/statistics). Let $\Sigma \subset \mathbb{R}^{3n}$ be the set of singularities of the underlying Coulomb potential. We show that the metric measure space $\mathscr{M}$ given by $\mathbb{R}^{3n}$ with its Euclidean distance and the measure \[ \mu (\mathrm{d}x)=e^{-2\Psi (x)}\mathrm{d}x \] has a Bakry-Emery-Ricci tensor which is absolutely bounded by the function $x\mapsto |x-\Sigma |^{-1}$, which we show to be an element of the Kato class induced by $\mathscr{M}$. In addition, it is shown that $\mathscr{M}$ is stochastically complete, that is, the Brownian motion which is induced by a molecule is nonexplosive. Our proofs reveal a fundamental connection between the above geometric/probabilistic properties and recently obtained derivative estimates for $\tilde{\Psi }$ by Fournais/Sørensen, as well as Aizenman/Simon’s Harnack inequality for Schrödinger operators. Moreover, our results suggest to study general metric measure spaces having a Ricci curvature which is synthetically bounded from below/above by a function in the underlying Kato class.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.76/ |
| spellingShingle | Güneysu, Batu von Renesse, Max Molecules as metric measure spaces with Kato-bounded Ricci curvature Comptes Rendus. Mathématique |
| title | Molecules as metric measure spaces with Kato-bounded Ricci curvature |
| title_full | Molecules as metric measure spaces with Kato-bounded Ricci curvature |
| title_fullStr | Molecules as metric measure spaces with Kato-bounded Ricci curvature |
| title_full_unstemmed | Molecules as metric measure spaces with Kato-bounded Ricci curvature |
| title_short | Molecules as metric measure spaces with Kato-bounded Ricci curvature |
| title_sort | molecules as metric measure spaces with kato bounded ricci curvature |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.76/ |
| work_keys_str_mv | AT guneysubatu moleculesasmetricmeasurespaceswithkatoboundedriccicurvature AT vonrenessemax moleculesasmetricmeasurespaceswithkatoboundedriccicurvature |