$\mathrm{RCD}(K,N)$ spaces and the geometry of multiparticle Schrödinger 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 selfadjoint 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 selfadjoint 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*} \bige^{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 multiparticle Schrödinger semigroups and, by the explicitness of the Hölder constants, sheds some light into the geometry of such operators.
 Publication:

arXiv eprints
 Pub Date:
 September 2019
 arXiv:
 arXiv:1909.07736
 Bibcode:
 2019arXiv190907736G
 Keywords:

 Mathematical Physics;
 Mathematics  Differential Geometry;
 Mathematics  Probability