1998/08/31 by Kurt Broderix, Dirk Hundertmark, Hajo Leschke
Physics and Astronomy · Mathematics · #math-ph #math.FA #math.MP #math.PR #msc:35J10
published as Published in slightly different form in Rev. Math. Phys. Vol. 12, pp 181--225 (2000) · Final Version, 51 pages, citation index, no figures
arxiv created 2000/03/16 · arxiv updated 2009/11/30
The objects of the present study are one-parameter semigroups generated by Schrödinger 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 Schrödinger 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.