$\mathrm{RCD}^*(K,N)$ spaces and the geometry of multi-particle Schr\"odinger semigroups

Abstract

With $(X,\mathfrak{d},\mathfrak{m})$ an $\mathrm{RCD}^*(K,N)$ space for some $K\in\mathbf{R}$, $N\in [1,\infty)$, 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 $V:X\to \mathbf{R}$ in this class, with $H_V$ the natural self-adjoint realization of the Schr\"odinger 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 $\alpha$-H\"older continuous. This result applies to multi-particle Schr\"odinger semigroups and, by the explicitness of the H\"older constants, sheds some light into the geometry of such operators.