Molecules as metric measure spaces with Kato-bounded Ricci curvature

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 $Σ\subset \IR^{3n}$ 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/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.

L^{\infty}-to-Lipschitz smoothing property of the heat flow added