Molecules as metric measure spaces with Kato-bounded Ricci curvature

Batu Güneysu,Max Von Renesse

doi:10.5802/crmath.76

Abstract

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 $\Sigma\subset \IR^{3n}$ be the set of singularities of the underlying Coulomb potential. We show that the metric measure space $\IMM$ given by $\IR^{3n}$ with its Euclidean distance and the measure $$ \mu(dx)=e^{-2\Psi(x)}dx $$ has a Bakry-Emery-Ricci tensor which is absolutely bounded by the the function $x\mapsto |x-\Sigma|^{-1}$, which we show to be an element of the Kato class induced by $\IMM$. In addition, it is shown $\IMM$ is stochastically complete, that is, the Brownian motion which is induced by a molecule is nonexplosive, and that the heat semigroup of $\IMM$ has the $L^{\infty}$-to-Lipschitz smoothing property. Our proofs reveal a fundamental connection between the above geometric/probabilistic properties and recently obtained derivative estimates for $e^{\Psi}$ by Fournais/Sorensen, as well as Aizenman/Simon's Harnack inequality for Schrodinger 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.

Highlights

We show that the metric measure space M given by R3n with its Euclidean distance and the measure μ(dx) = e−2Ψ(x)dx has a Bakry-Emery-Ricci tensor which is absolutely bounded by the function x → |x − Σ|−1, which we show to be an element of the Kato class induced by M
Ever since the pioneering papers by Sturm [25,26] and Lott/Villani [19], which based on the earlier results from [9, 21, 22], metric measure spaces with a Ricci curvature which is bounded below by a constant have been examined in great detail and revealed many deep geometric and analytic results and in particular stability properties
The reader may find some of the central results on the geometry and analysis of abstract metric measures with lower bounded Ricci curvature in [2,3,4,7,8,10,12,17,20] and the references therein

Summary

Introduction

Ever since the pioneering papers by Sturm [25,26] and Lott/Villani [19], which based on the earlier results from [9, 21, 22], metric measure spaces with a Ricci curvature which is bounded below by a constant have been examined in great detail and revealed many deep geometric and analytic results and in particular stability properties. We show that the metric measure space M given by R3n with its Euclidean distance and the measure μ(dx) = e−2Ψ(x)dx has a Bakry-Emery-Ricci tensor which is absolutely bounded by the function x → |x − Σ|−1, which we show to be an element of the Kato class induced by M .

Results

Conclusion