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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 Laplacianinduced by the underlying Cheeger form. Given $\alpha\in [0,1]$ we introducethe $\alpha$-Kato class of potentials on $(X,\mathfrak{d},\mathfrak{m})$, andgiven a potential $V:X\to \mathbf{R}$ in this class, with $H_V$ the naturalself-adjoint realization of the Schr\"odinger operator $H+V$ in$L^2(X,\mathfrak{m})$, we use Brownian coupling methods and perturbation theoryto prove that for all $t>0$ there exists an explicitly given constant$A(V,K,\alpha,t)<\infty$, such that for all $\Psi\inL^{\infty}(X,\mathfrak{m})$, $x,y\in X$ 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\"oldercontinuous. This result applies to multi-particle Schr\"odinger semigroups and,by the explicitness of the H\"older constants, sheds some light into thegeometry of such operators.

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