Molecules as metric measure spaces with Kato-bounded Ricci curvature

Güneysu, Batu; von Renesse, Max

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

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

$$\mu \left(\mathrm{d}x\right)=e^{-2\Psi \left(x\right)}\mathrm{d}x$$

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 $ℳ$. 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 $\tilde{\Psi }$ 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.

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