Communications in Mathematical Physics

, Volume 343, Issue 1, pp 311–359 | Cite as

Long Time Quantum Evolution of Observables on Cusp Manifolds

  • Yannick Bonthonneau


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 \({|\mathfrak{I}s| \to \infty}\) and \({\mathfrak{R}s \to d/2}\) with \({\mathfrak{R}s - d/2 \geq \log \log |\mathfrak{I}s| / \log |\mathfrak{I}s|}\). 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.


Manifold Covariant Derivative Pseudodifferential Operator Eisenstein Series Resonant State 
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© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.UQÀM, CIRGETQuébecCanada

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