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Original Paper

Compactness of Schrödinger semigroups

  1. Daniel Lenz1,†,
  2. Peter Stollmann2,‡,
  3. Daniel Wingert2,*

Article first published online: 23 DEC 2009

DOI: 10.1002/mana.200910054

Issue

Mathematische Nachrichten

Mathematische Nachrichten

Special Issue: Erhard Schmidt Memorial Issue, Part I

Volume 283, Issue 1, pages 94–103, January 2010

Additional Information

How to Cite

Lenz, D., Stollmann, P. and Wingert, D. (2010), Compactness of Schrödinger semigroups. Math. Nachr., 283: 94–103. doi: 10.1002/mana.200910054

Author Information

  1. 1

    Mathematisches Institut, Friedrich-Schiller Universität Jena, Ernst-Abbé Platz 2, 07743 Jena, Germany

  2. 2

    Fakultät für Mathematik, Technische Universität, 09107 Chemnitz, Germany

  1. †

    Phone: +49 (0)3641 946131, Fax: +49 (0)3641 946102

  2. ‡

    +49 (0)371 53138349, Fax: +49 (0)371 53122209

*Phone: +49 (0)371 53137641, Fax: +49 (0)371 53122209

  1. †

    Phone: +49 (0)3641 946131, Fax: +49 (0)3641 946102

  2. ‡

    +49 (0)371 53138349, Fax: +49 (0)371 53122209

Publication History

  1. Issue published online: 23 DEC 2009
  2. Article first published online: 23 DEC 2009
  3. Manuscript Accepted: 20 MAR 2009
  4. Manuscript Received: 2 MAR 2009

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Keywords:

  • Compact resolvent;
  • Schrödinger operators

Abstract

This paper is concerned with emptyness of the essential spectrum, or equivalently compactness of the semigroup, for perturbations of self-adjoint operators that are bounded below (on an L2-space).

For perturbations by a (nonnegative) potential we obtain a simple criterion for compactness of the semigroup in terms of relative compactness of the operators of multiplication with characteristic functions of sublevel sets. In the context of Dirichlet forms, we can even characterize compactness of the semigroup for measure perturbations. Here, certain ‘averages’ of the measure outside of compact sets play a role.

As an application we obtain compactness of semigroups for Schrödinger operators with potentials whose sublevel sets are thin at infinity (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

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