RCD*(K,N) Spaces and the Geometry of Multi-Particle Schrödinger Semigroups

Dedicated to the memory of Kazumasa Kuwada. Let $(X,\mathfrak{d},{\mathfrak{m}})$ be an $\textrm{RCD}^*(K,N)$ space for some $K\in{\mathbb{R}}$, $N\in [1,\infty )$, and let $H$ be the self-adjoint Laplacian induced by the underlying Cheeger form. Given $\alpha \in [0,1]$, we introduce the $\alpha$-Kato class of potentials on $(X,\mathfrak{d},{\mathfrak{m}})$, and given a potential $V:X\to{\mathbb{R}}$ in this class, we denote with $H_V$ the natural self-adjoint realization of the Schrödinger operator $H+V$ in $L^2(X,{\mathfrak{m}})$. We use Brownian coupling methods and perturbation theory to prove that for all $t>0$, there exists an explicitly given constant $A(V,K,\alpha ,t)<\infty$, such that for all $\Psi \in L^{\infty }(X,{\mathfrak{m}})$, $x,y\in X$ one has $$\begin{align*}\big|e^{-tH_V}\Psi(x)-e^{-tH_V}\Psi(y)\big|\leq A(V,K,\alpha,t) \|\Psi\|_{L^{\infty}}\mathfrak{d}(x,y)^{\alpha}.\end{align*}$$In particular, all $L^{\infty }$-eigenfunctions of $H_V$ are globally $\alpha$-Hölder continuous. This result applies to multi-particle Schrödinger semigroups and, by the explicitness of the Hölder constants, sheds some light into the geometry of such operators.