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High-energy limits of Laplace-type and Dirac-type eigenfunctions and frame flows


Authors: Dmitry Jakobson and Alexander Strohmaier
Journal: Electron. Res. Announc. Amer. Math. Soc. 12 (2006), 87-94
MSC (2000): Primary 81Q50; Secondary 35P20, 37D30, 58J50, 81Q005
DOI: https://doi.org/10.1090/S1079-6762-06-00164-8
Published electronically: July 25, 2006
MathSciNet review: 2237272
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Abstract: We relate high-energy limits of Laplace-type and Dirac-type operators to frame flows on the corresponding manifolds, and show that the ergodicity of frame flows implies quantum ergodicity in an appropriate sense for those operators. Observables for the corresponding quantum systems are matrix-valued pseudodifferential operators, and therefore the system remains noncommutative in the high-energy limit. We discuss to what extent the space of stationary high-energy states behaves classically.


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Additional Information

Dmitry Jakobson
Affiliation: Department of Mathematics and Statistics, McGill University, 805 Sherbrooke St. West, Montréal QC H3A 2K6, Canada
Email: jakobson@math.mcgill.ca

Alexander Strohmaier
Affiliation: Mathematisches Institut, Universität Bonn, Beringstrasse 1, D-53115 Bonn, Germany
Email: strohmai@math.uni-bonn.de

DOI: https://doi.org/10.1090/S1079-6762-06-00164-8
Keywords: Dirac operator, Hodge Laplacian, eigenfunction, frame flow, quantum ergodicity
Received by editor(s): April 26, 2006
Published electronically: July 25, 2006
Additional Notes: The first author was supported by NSERC, FQRNT and Dawson fellowship.
Communicated by: S. Katok
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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