Annales Henri Poincaré

, Volume 16, Issue 1, pp 1–14 | Cite as

Quantum Ergodicity for a Point Scatterer on the Three-Dimensional Torus

  • Nadav YeshaEmail author


Consider a point scatterer (the Laplacian perturbed by a delta-potential) on the standard three-dimensional flat torus. Together with the eigenfunctions of the Laplacian which vanish at the point, this operator has a set of new, perturbed eigenfunctions. In a recent paper, the author was able to show that all of the perturbed eigenfunctions are uniformly distributed in configuration space. In this paper we prove that almost all of these eigenfunctions are uniformly distributed in phase space, i.e. we prove quantum ergodicity for the subspace of the perturbed eigenfunctions. An analogue result for a point scatterer on the two-dimensional torus was recently proved by Kurlberg and Ueberschär.


Point Scatterer Unperturbed Problem Quantum Ergodicity Integer Lattice Point Standard Basis Element 
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.


  1. 1.
    Agranovich, M.S.: Spectral properties of elliptic pseudodifferential operators on a closed curve. (Russian) Funktsional. Anal. i Prilozhen. 13(4), 54–56 (1979). [English translation in functional analysis and its applications. 13, 279–281 (1979)]Google Scholar
  2. 2.
    Atkinson, K., Han, W.: Spherical harmonics and approximations on the unit sphere: an introduction. Springer, Berlin (2012)Google Scholar
  3. 3.
    Colin de Verdière Y.: Pseudo-laplaciens I. Annales de l’Institut Fourier 32(3), 275–286 (1982)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Colin de Verdière Y.: Ergodicité et fonctions propres du laplacien. Comm. Math. Phys. 102, 497–502 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Duke W.: Hyperbolic distribution problems and half-integral weight Maass forms. Invent. Math. 92, 73–90 (1988)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Duke W., Schulze-Pillot R.: Representation of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids. Invent. Math. 99, 49–57 (1990)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Erdös P., Hall R.R.: On the angular distribution of Gaussian integers with fixed norm. Discr. Math. 200, 87–94 (1999)CrossRefzbMATHGoogle Scholar
  8. 8.
    Golubeva, E.P., Fomenko, O.M.: Asymptotic distribution of integral points on the three-dimensional sphere. Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst. Akad. Nauk SSSR 160, 54–71 (1987). [English translation in journal of soviet mathematics 52(3), 3036–3048 (1990)]Google Scholar
  9. 9.
    Golubeva, E.P., Fomenko, O.M.: Remark on asymptotic distribution of the integral points on the large three-dimensional sphere. Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst. Akad. Nauk SSSR 185, 22–28 (1990). [English translation in journal of soviet mathematics 59(6), 1148–1152 (1992)]Google Scholar
  10. 10.
    Grosswald E.: Representations of integers as sums of squares. Springer, New York (1985)CrossRefzbMATHGoogle Scholar
  11. 11.
    McLean W.: Local and global descriptions of periodic pseudodifferential operators. Math. Nachr. 150, 151–161 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Rudnick Z., Ueberschär H.: Wave function statistics for a point scatterer on the torus. Comm. Math. Phys. 316, 763–782 (2012)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Ruzhansky, M., Turunen, V.: Pseudo-differential operators and symmetries: background analysis and advanced topics. Birkhäuser, Basel (2010)Google Scholar
  14. 14.
    Schnirelman A.: Ergodic properties of eigenfunctions. Usp. Math. Nauk. 29, 181–182 (1974)Google Scholar
  15. 15.
    Shigehara T.: Conditions for the appearance of wave chaos in quantum singular systems with a pointlike scatterer. Phys. Rev. E 50, 4357–4370 (1994)ADSCrossRefGoogle Scholar
  16. 16.
    Shigehara, T., Cheon, T.: Spectral properties of three-dimensional quantum billiards with a pointlike scatterer. Phys. Rev. E 55, 6832–6844 (1997)Google Scholar
  17. 17.
    Siegel C.L.: Uber die Classenzahl quadratischer Zahlkörper. Acta Arith. 1, 83–86 (1935)Google Scholar
  18. 18.
    Yesha N.: Eigenfunction statistics for a point-scatterer on a three-dimensional torus. Annales Henri Poincaré 14(7), 1801–1836 (2013)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Zelditch S.: Uniform distribution of eigenfunctions on compact hyperbolic surfaces. Duke Math. J. 55, 919–941 (1987)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Raymond and Beverly Sackler School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael

Personalised recommendations