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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Note on quantum unique ergodicity


Author: Steve Zelditch
Journal: Proc. Amer. Math. Soc. 132 (2004), 1869-1872
MSC (2000): Primary 58J50, 58J40, 35P99, 81S10
Published electronically: November 21, 2003
MathSciNet review: 2051153
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Abstract: We prove that (near) off-diagonal matrix elements $\langle A \varphi_i, \varphi_j \rangle$ ($i \not= j$) of pseudodifferential operators relative to eigenfunctions of quantum unique- ly ergodic Laplacians vanish as the eigenvalues tend to infinity. It follows that QUE systems cannot have quasi-modes with singular limits and a bounded number of essential frequencies, as is believed to occur for the stadium and other examples.


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Additional Information

Steve Zelditch
Affiliation: Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218
Email: zelditch@math.jhu.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-03-07298-8
PII: S 0002-9939(03)07298-8
Received by editor(s): January 28, 2003
Received by editor(s) in revised form: March 10, 2003
Published electronically: November 21, 2003
Additional Notes: This research was partially supported by NSF grant DMS-0071358 and by the Clay Mathematics Institute
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 2003 American Mathematical Society