Long Time Quantum Evolution of Observables on Cusp Manifolds

The Eisenstein functions E ( s ) are some generalized eigenfunctions of the Laplacian on manifolds with cusps. We give a version of Quantum Unique Ergodicity for them, for | I s | → ∞ and R s → d / 2 with R s - d / 2 ≥ log log | I s | / log | I s | . For the purpose of the proof, we build a semi-cla...

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Bibliographic Details
Published in:Communications in mathematical physics 2016-04, Vol.343 (1), p.311-359
Main Author: Bonthonneau, Yannick
Format: Article
Language:English
Subjects:
Classical and Quantum Gravitation
Complex Systems
Mathematical and Computational Physics
Mathematical Physics
Physics
Physics and Astronomy
Quantum Physics
Relativity Theory
Theoretical
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Summary:The Eisenstein functions E ( s ) are some generalized eigenfunctions of the Laplacian on manifolds with cusps. We give a version of Quantum Unique Ergodicity for them, for | I s | → ∞ and R s → d / 2 with R s - d / 2 ≥ log log | I s | / log | I s | . For the purpose of the proof, we build a semi-classical quantization procedure for finite volume manifolds with hyperbolic cusps, adapted to a geometrical class of symbols. We also prove an Egorov Lemma until Ehrenfest times on such manifolds.
ISSN:0010-3616
1432-0916
DOI:10.1007/s00220-016-2573-3