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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Note on converse quantum ergodicity
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by Boris Gutkin PDF
Proc. Amer. Math. Soc. 137 (2009), 2795-2800 Request permission

Abstract:

Quantum ergodicity asserts that eigenstates of a system with classical ergodic dynamics must be “equidistributed” in the phase space. In the present note we show that the converse is not true. We provide an example of billiards which are quantum ergodic but not classically ergodic.
References
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Additional Information
  • Boris Gutkin
  • Affiliation: Fachbereich Physik, Universität Duisburg-Essen, 47048 Duisburg, Germany
  • Email: boris.gutkin@uni-duisburg-essen.de
  • Received by editor(s): August 12, 2008
  • Received by editor(s) in revised form: December 8, 2008
  • Published electronically: March 5, 2009
  • Additional Notes: The author would like to thank S. Nonnenmacher for helpful discussions on the converse quantum ergodicity problem and A. Knauf for a careful reading of the manuscript and valuable comments. The financial support of the Minerva Foundation and SFB/TR12 of the Deutsche Forschungsgemainschaft is acknowledged.
  • Communicated by: Bryna Kra
  • © Copyright 2009 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 137 (2009), 2795-2800
  • MSC (2000): Primary 58J50, 81Q50; Secondary 37D50
  • DOI: https://doi.org/10.1090/S0002-9939-09-09849-9
  • MathSciNet review: 2497494