H\"older estimates for magnetic Schr\"odinger 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 Schr\"odinger semigroup in $\mathbb{R}$ has a global $L^{p}$-to-$C^{0,\beta}$ H\"older smoothing property for all $p\in [1,\infty]$, in particular all eigenfunctions are uniformly $\beta$-H\"older continuous. This result shows that the eigenfunctions of the Hamilton operator of a molecule in a magnetic field are uniformly $\beta$-H\"older continuous under weak $L^q$-assumptions on the magnetic potential.