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Continuity properties of Schrödinger semigroups with magnetic fields

Kurt Broderix·Dirk Hundertmark·Hajo Leschke
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Abstract

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.

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Kurt Broderix, Dirk Hundertmark, Hajo Leschke. 2000-03-16. Continuity properties of Schrödinger semigroups with magnetic fields. https://arxiv.org/abs/math-ph/9808004

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