Communications in Mathematical Physics

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

Long Time Quantum Evolution of Observables on Cusp Manifolds

Article

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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References

  1. Bab02.
    Babillot M.: On the mixing property for hyperbolic systems. Israel J. Math. 129, 61–76 (2002)MathSciNetCrossRefMATHGoogle Scholar
  2. Bou14.
    Bouclet, J.-M.: Strichartz inequalities on surfaces with cusps. Int. Math. Res. Notices (2016, to appear). arXiv:1405.2123
  3. BR02.
    Bouzouina A., Robert D.: Uniform semiclassical estimates for the propagation of quantum observables. Duke Math. J. 111(2), 223–252 (2002)MathSciNetCrossRefMATHGoogle Scholar
  4. CdV81.
    de Verdière Yves, C.: Une nouvelle démonstration du prolongement méromorphe des séries d’Eisenstein. C. R. Acad. Sci. Paris Sér. I Math. 293(7):361–363 (1981)Google Scholar
  5. CE75.
    Cheeger, J., Ebin, D.G.: Comparison Theorems in Riemannian Geometry, North-Holland Mathematical Library, vol. 9. North-Holland, Amsterdam (1975)Google Scholar
  6. Cou07.
    Coudène Y.: The Hopf argument. J. Modern Dyn. 1(1), 147–153 (2007)MathSciNetCrossRefMATHGoogle Scholar
  7. DG14.
    Dyatlov S., Guillarmou C.: Microlocal limits of plane waves and Eisenstein functions. Ann. Sci. Éc. Norm. Supér. (4) 47(2), 371–448 (2014)MathSciNetMATHGoogle Scholar
  8. DS99.
    Dimassi M., Sjöstrand J.: Spectral Asymptotics in the Semi-classical Limit, London Mathematical Society Lecture Note Series, vol. 268. Cambridge University Press, Cambridge (1999)CrossRefMATHGoogle Scholar
  9. Dya12.
    Dyatlov S.: Microlocal limits of Eisenstein functions away from the unitarity axis. J. Spectr. Theory 2(2), 181–202 (2012)MathSciNetCrossRefMATHGoogle Scholar
  10. GK02.
    Gudmundsson S., Kappos E.: On the geometry of tangent bundles. Expo Math. 20(1), 1–41 (2002)MathSciNetCrossRefMATHGoogle Scholar
  11. GN14.
    Guillarmou C., Naud F.: Equidistribution of Eisenstein series on convex co-compact hyperbolic manifolds. Am. J. Math. 136(2), 445–479 (2014)MathSciNetCrossRefMATHGoogle Scholar
  12. Hör03.
    Hörmander, L.: The analysis of linear partial differential operators. I. Classics in Mathematics. Springer, Berlin, 2003 (Distribution theory and Fourier analysis, Reprint of the second (1990) edition. Springer, Berlin)Google Scholar
  13. Jak94.
    Jakobson, D. Quantum unique ergodicity for Eisenstein series on \({\rm PSL}_2({\bf Z})\backslash {\rm PSL}_2({\bf R})\). Ann. Inst. Fourier (Grenoble) 44(5),1477–1504 (1994)MathSciNetCrossRefGoogle Scholar
  14. LP76.
    Lax P.D., Phillips R.S.: Scattering Theory for Automorphic Functions, Annals of Mathematics Studies, vol. 87. Princeton University Press, Princeton (1976)Google Scholar
  15. LS95.
    Luo, W.Z., Sarnak, P.: Quantum ergodicity of eigenfunctions on \({{\rm PSL}_2(Z)\backslash H^2}\). Inst. Hautes Études Sci. Publ. Math. 81, 207–237 (1995)Google Scholar
  16. MM98.
    Mazzeo, R., Melrose, R.B.: Pseudodifferential operators on manifolds with fibred boundaries. Asian J. Math. 2(4), 833–866 (1998) (Mikio Sato: a great Japanese mathematician of the twentieth century)Google Scholar
  17. Mül83.
    Müller W.: Spectral theory for Riemannian manifolds with cusps and a related trace formula. Math. Nachr. 111, 197–288 (1983)MathSciNetCrossRefGoogle Scholar
  18. Mül92.
    Müller W.: Spectral geometry and scattering theory for certain complete surfaces of finite volume. Invent. Math. 109(2), 265–305 (1992)ADSMathSciNetCrossRefMATHGoogle Scholar
  19. Pau14.
    Paulin, F.: Géométrie riemannienne. Lecture notes (2014). http://www.math.u-psud.fr/~paulin/notescours/cours_georiem (page 205, equation 40)
  20. PPS15.
    Paulin, F., Pollicott, M., Schapira, B.: Equilibrium states in negative curvature. Astérisque 373, Soc. Math. (2015). arXiv:1211.6242
  21. PRR13.
    Petridis, Y.N., Raulf, N., Risager, M.S.: Quantum limits of eisenstein series and scattering states. Canad. Math. Bull. 56, 814–826, 827–828 (2013)Google Scholar
  22. Tay11.
    Taylor M.E.: Partial Differential Equations I. Basic Theory, Applied Mathematical Sciences, vol. 115, 2nd edn. Springer, New York (2011)Google Scholar
  23. Zel86.
    Zelditch S.: Pseudodifferential analysis on hyperbolic surfaces. J. Funct. Anal. 68(1), 72–105 (1986)MathSciNetCrossRefMATHGoogle Scholar
  24. Zel91.
    Zelditch S.: Mean Lindelöf hypothesis and equidistribution of cusp forms and Eisenstein series. J. Funct. Anal. 97(1), 1–49 (1991)MathSciNetCrossRefMATHGoogle Scholar
  25. Zwo12.
    Zworski M.: Semiclassical Analysis, Graduate Studies in Mathematics, vol. 138. American Mathematical Society, Providence (2012)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.UQÀM, CIRGETQuébecCanada

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