Abstract
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.
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Bonthonneau, Y. Long Time Quantum Evolution of Observables on Cusp Manifolds. Commun. Math. Phys. 343, 311–359 (2016). https://doi.org/10.1007/s00220-016-2573-3
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DOI: https://doi.org/10.1007/s00220-016-2573-3