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

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

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