Continuity properties of Schrödinger semigroups with magnetic fields
Abstract
The objects of the present study are oneparameter 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 Katolike conditions. The configuration space is supposed to be an arbitrary open subset of multidimensional 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 localnormcontinuity of the semigroup in the potentials. Finally, we prove that the semigroup has a continuous integral kernel given by a Brownianbridge 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}, 447526 (1982)] to nonzero vector potentials and more general configuration spaces.
 Publication:

arXiv eprints
 Pub Date:
 August 1998
 DOI:
 10.48550/arXiv.mathph/9808004
 arXiv:
 arXiv:mathph/9808004
 Bibcode:
 1998math.ph...8004B
 Keywords:

 Mathematical Physics;
 Mathematics  Functional Analysis;
 Mathematics  Mathematical Physics;
 Mathematics  Probability;
 35J10
 EPrint:
 Final Version, 51 pages, citation index, no figures