Proceedings of the American Mathematical Society

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



Note on converse quantum ergodicity

Author: Boris Gutkin
Journal: Proc. Amer. Math. Soc. 137 (2009), 2795-2800
MSC (2000): Primary 58J50, 81Q50; Secondary 37D50
Published electronically: March 5, 2009
MathSciNet review: 2497494
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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.

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Additional Information

Boris Gutkin
Affiliation: Fachbereich Physik, Universität Duisburg-Essen, 47048 Duisburg, Germany

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
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.