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.

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RCD*(K,N) 空间和多粒子薛定谔半群的几何

用 $(X,\mathfrak{d},\mathfrak{m})$ 一个 $\mathrm{RCD}(K,N)$ 空间用于一些 $K\in\mathbf{R}$, $N\in [ 1,\infty)$，令$H$ 是由底层 Cheeger 形式诱导的自伴随拉普拉斯算子。给定 $\alpha\in [0,1]$，我们在 $(X,\mathfrak{d},\mathfrak{m})$ 上引入 $\alpha$-Kato 类势，并给定一个势 $V： X\to \mathbf{R}$ 在这个类中，$H_V$ 是 Schr\"odinger 算子 $H+V$ 在 $L^2(X,\mathfrak{m})$ 中的自然自伴随实现，我们使用布朗耦合方法和微扰理论来证明对于所有 $t>0$ 存在一个明确给定的常数 $A(V,K,\alpha,t)<\infty$，使得对于所有 $\Psi\在 L^{\infty}(X,\mathfrak{m})$, $x,y\in X$ 中有 \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}。