Hölder estimates for magnetic Schrödinger semigroups in ${\mathbb{R}}^d$ from mirror coupling

Abstract

We use the mirror coupling of Brownian motion to show that under a $\beta \in (0,1)$-dependent Kato type assumption (which is satisfied under a suitable $L^q$-assumption on the electro-magnetic potential, where $q$ depends on $\beta$ and the dimension $d$) on the possibly nonsmooth electro-magnetic potential, the corresponding magnetic Schrödinger semigroup in $\mathbb{R}$ has a global $L^p$-to-$C^{0,\beta }$ Hölder smoothing property for all $p\in [1,\infty ]$, in particular all eigenfunctions are uniformly $\beta$-Hölder continuous. This result shows that the eigenfunctions of the Hamilton operator of a molecule in a magnetic field are uniformly $\beta$-Hölder continuous under weak $L^q$-assumptions on the magnetic potential.

Published version (in Letters of Mathematical Physics) of this paper. Corrected placement of footnote 1 (was misplaced in the published version in the production phase)