Abstract
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.
Perturbation theory for the periodic multidimensional Schrodinger operator and the Bethe-Sommerfeld conjecture
