Molecules as metric measure spaces with Kato-bounded Ricci curvature

Batu Güneysu; Max von Renesse

doi:10.5802/crmath.76

Probabilités, Physique mathématique

Molecules as metric measure spaces with Kato-bounded Ricci curvature

Batu Güneysu ¹ ; Max von Renesse ²

¹ Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany
² Fakultät für Mathematik und Informatik, Universität Leipzig, Ritterstraße 26, 04109 Leipzig, Germany

Comptes Rendus. Mathématique, Volume 358 (2020) no. 5, pp. 595-602.

Résumé

Set $Ψ : = log (\tilde{Ψ})$ , with $\tilde{Ψ} > 0$ the ground state of an arbitrary molecule with $n$ electrons in the infinite mass limit (neglecting spin/statistics). Let $Σ \subset ℝ^{3 n}$ be the set of singularities of the underlying Coulomb potential. We show that the metric measure space $ℳ$ given by $ℝ^{3 n}$ with its Euclidean distance and the measure

μ (d x) = e^{- 2 Ψ (x)} d x

has a Bakry-Emery-Ricci tensor which is absolutely bounded by the function $x \mapsto {| x - Σ |}^{- 1}$ , which we show to be an element of the Kato class induced by $ℳ$ . In addition, it is shown that $ℳ$ 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 $\tilde{Ψ}$ 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.

Reçu le : 2019-09-11
Révisé le : 2020-04-30
Accepté le : 2020-05-19
Publié le : 2020-09-14

DOI : 10.5802/crmath.76

Affiliations des auteurs :

Batu Güneysu ¹ ; Max von Renesse ²

¹ Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany
² Fakultät für Mathematik und Informatik, Universität Leipzig, Ritterstraße 26, 04109 Leipzig, Germany

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Batu G\"uneysu and Max von Renesse},
     title = {Molecules as metric measure spaces with {Kato-bounded} {Ricci} curvature},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {595--602},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {358},
     number = {5},
     year = {2020},
     doi = {10.5802/crmath.76},
     language = {en},
}

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Batu Güneysu; Max von Renesse. Molecules as metric measure spaces with Kato-bounded Ricci curvature. Comptes Rendus. Mathématique, Volume 358 (2020) no. 5, pp. 595-602. doi : 10.5802/crmath.76. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.76/

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