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Note on converse quantum ergodicity


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