Communications in Mathematical Physics

, Volume 161, Issue 1, pp 195–213 | Cite as

The behaviour of eigenstates of arithmetic hyperbolic manifolds

  • Zeév Rudnick
  • Peter Sarnak


In this paper we study some problems arising from the theory of Quantum Chaos, in the context of arithmetic hyperbolic manifolds. We show that there is no strong localization (“scarring”) onto totally geodesic submanifolds. Arithmetic examples are given, which show that the random wave model for eigenstates does not apply universally in 3 degrees of freedom.


Neural Network Manifold Statistical Physic Complex System Nonlinear Dynamics 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arnold, V., Avez, A.: Ergodic problems of classical mechanics. New York: Benjamin 1968Google Scholar
  2. 2.
    Aurich, R., Steiner, F.: Energy level statistics of the Hadamard-Gutzwiller ensemble. Physica D43, 155–180 (1990)Google Scholar
  3. 3.
    Aurich, R., Steiner, F.: Statistical properties of highly excited quantum eigenstates of a strongly chaotic system. Preprint DESY 92-091, June 1992Google Scholar
  4. 4.
    Bogomolny, E.B., Georgeot, B., Giannoni, M., Schmidt, C.: Chaotic billiards generated by arithmetic groups. Phys. Rev. Lett.69, 1477–1480 (1992)CrossRefGoogle Scholar
  5. 5.
    Cassels, J.W.: Rational quadratic forms. New York: Academic Press (1978)Google Scholar
  6. 6.
    Colin de Verdiere, Y.: Ergodicité et functions propre du laplacien. Commun. Math. Phys.102, 497–502 (1985)CrossRefGoogle Scholar
  7. 7.
    Conway, J., Sloane, N.: Sphere packings, lattices and groups. Berlin, Heidelberg, New York: Springer 1988Google Scholar
  8. 8.
    Eichler, M.: Lectures on modular correspondences. Tata Institute9, 1955Google Scholar
  9. 9.
    Hejhal, D., Rackner, B.: On the topography of Maass waveforms forPSL(2,Z); Experiments and heuristics. Experimental Math.1, 275–306 (1992)Google Scholar
  10. 10.
    Helgason, S.: Groups and Geometric Analysis. New York: Academic Press (1984)Google Scholar
  11. 11.
    Heller, E.J.: In: Chaos and Quantum Phyiscs, Les Houches 1989 (ed. by M.J. Giannoni, A. Voros, and J. Zinn-Justin), Amsterdam: North-Holland, 1991, pp. 549–661Google Scholar
  12. 12.
    Hormander, L.: The Analysis of Linear Partial Differential Operators, Vol.I–IV, Berlin, Heidelberg, New York: Springer-Verlag, 1985Google Scholar
  13. 13.
    Howe, R., Piatetski-Shapiro, I.: A counter-example to the “generalized Ramanujan conjecture” for (quasi-) split groups. Proc. Symp. in Pure Math. vol.33, Amer. Math. Soc. 315–322 (1979)Google Scholar
  14. 14.
    Iwaniec, H., Sarnak, P.:L norms of eigenfunctions of arithmetic surfaces. PreprintGoogle Scholar
  15. 15.
    Landau, E.: Elementary Number Theory. New York: Chelsea Pub. Co., 1958Google Scholar
  16. 16.
    Luo, W., Sarnak, P.: Number variance for arithmetic hyperbolic surfaces. To appear in Commun. Math. Phys.Google Scholar
  17. 17.
    Maass, H.: Über die räumliche Verteilung der Punkte in Gittern mit indefiniter Metrik. Math. Annalen138, 287–315 (1959)CrossRefGoogle Scholar
  18. 18.
    Sarnak, P.: Arithmetic Quantum Chaos. Schur Lectures, Tel Aviv 1992, preprintGoogle Scholar
  19. 19.
    Schnirelman, A.: Usp. Mat. Nauk29, 181–182 (1974)Google Scholar
  20. 20.
    Selberg, A.: Gottingen Lectures. In: Collected Works, vol.1, Berlin, Heidelberg, New York: Springer-VerlagGoogle Scholar
  21. 21.
    Shintani, T.: On construction of holomorphic forms of half integral weight. Nagoya Math. J.58, 83–126 (1975)Google Scholar
  22. 22.
    Seeger, A., Sogge, C.D.: Bounds for eigenfunctions of differential operators. Indiana University Math. J.38, 669–682 (1989)CrossRefGoogle Scholar
  23. 23.
    Zelditch, S.: Uniform distribution of eigenfunctions on compact hyperbolic surfaces. Duke Math. J.55, 919–941 (1987)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • Zeév Rudnick
    • 1
  • Peter Sarnak
    • 1
  1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA

Personalised recommendations