Long Time Quantum Evolution of Observables on Cusp Manifolds

The Eisenstein functions E(s)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${E(s)}$$\end{document} are some generalized eigenfunctions of the Laplacian on manifolds with cusps. We give a version of Quantum Unique Ergodicity for them, for |Is|→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${|\mathfrak{I}s| \to \infty}$$\end{document} and Rs→d/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak{R}s \to d/2}$$\end{document} with Rs-d/2≥loglog|Is|/log|Is|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak{R}s - d/2 \geq \log \log |\mathfrak{I}s| / \log |\mathfrak{I}s|}$$\end{document}. 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.