Continuity properties of Schr\"odinger semigroups with magnetic fields

TL;DR

This paper investigates the continuity properties of Schr"odinger semigroups with magnetic fields, extending classical results to more general electromagnetic potentials and configuration spaces, and establishing the existence of continuous integral kernels.

Contribution

It extends continuity and kernel results for Schr"odinger semigroups to include magnetic fields and broader spaces, generalizing Simon's foundational work.

Findings

Proved local-norm-continuity of the semigroup in potentials

Established the existence of a continuous integral kernel via Brownian-bridge expectations

Extended classical results to non-zero vector potentials and general spaces

Abstract

The objects of the present study are one-parameter semigroups generated by Schr\"odinger operators with fairly general electromagnetic potentials. More precisely, we allow scalar potentials from the Kato class and impose on the vector potentials only local Kato-like conditions. The configuration space is supposed to be an arbitrary open subset of multi-dimensional Euclidean space; in case that it is a proper subset, the Schr\"odinger operator is rendered symmetric by imposing Dirichlet boundary conditions. We discuss the continuity of the image functions of the semigroup and show local-norm-continuity of the semigroup in the potentials. Finally, we prove that the semigroup has a continuous integral kernel given by a Brownian-bridge expectation. Altogether, the article is meant to extend some of the results in B. Simon's landmark paper [Bull. Amer. Math. Soc. (N.S.) {\bf 7}, 447--526 (1982)] to non-zero vector potentials and more general configuration spaces.