${\mathrm{RCD}}^{\ast }(K,N)$ spaces and the geometry of multi-particle Schrödinger semigroups

Abstract

With $(X,\mathfrak{d},\mathfrak{m})$ an ${\mathrm{RCD}}^{\ast }(K,N)$ 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 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}Ψ(x)-e^{-tH_V}Ψ(y)\big|\leq A(V,K,α,t) \|Ψ\|_{L^{\infty}}\mathfrak{d}(x,y)^α. \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.

Some minor corrections; To appear in International Mathematics Research Notices