Skip to main content
SpringerLink
Log in

Statistics of Wave Functions for a Point Scatterer on the Torus

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

Quantum systems whose classical counterpart have ergodic dynamics are quantum ergodic in the sense that almost all eigenstates are uniformly distributed in phase space. In contrast, when the classical dynamics is integrable, there is concentration of eigenfunctions on invariant structures in phase space. In this paper we study eigenfunction statistics for the Laplacian perturbed by a delta-potential (also known as a point scatterer) on a flat torus, a popular model used to study the transition between integrability and chaos in quantum mechanics. The eigenfunctions of this operator consist of eigenfunctions of the Laplacian which vanish at the scatterer, and new, or perturbed, eigenfunctions. We show that almost all of the perturbed eigenfunctions are uniformly distributed in configuration space.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bambah R.P., Chowla S.: On numbers which can be expressed as a sum of two squares. Proc. Nat. Inst. Sci. India. 13, 101–103 (1947)

    MathSciNet  Google Scholar 

  2. Berkolaiko G., Keating J.P., Winn B.: Intermediate wave functions statistics. Phys. Rev. Lett. 91, 134103 (2003)

    Article  ADS  Google Scholar 

  3. Bogomolny, E., Gerland, U., Schmit, C.: Singular statistics. Phys. Rev. E (3) 63(3), part 2, 036206 (2001)

    Google Scholar 

  4. Bogomolny, E., Giraud, O., Schmit C.: Nearest-neighbor distribution for singular billiards. Phys. Rev. E (3) 65(5), 056214 (2002)

    Google Scholar 

  5. Shigehara T., Cheon T.: Wave chaos in quantum billiards with a small but finite-size scatterer. Phys. Rev. E 54, 1321–1331 (1996)

    Article  ADS  Google Scholar 

  6. Colin de Verdière Y.: Pseudo-laplaciens. I. Ann. l’Inst.Fourier 32(3), 275–286 (1982)

    Article  Google Scholar 

  7. Colinde Verdière Y.: Ergodicité et fonctions propres du laplacien. Commun. Math. Phys. 102(3), 497–502 (1985)

    Article  Google Scholar 

  8. Hejhal, D.A.: Some observations concerning eigenvalues of the Laplacian and Dirichlet L-series. In: Recent progress in analytic number theory, Symp. Durham 1979, Vol. 2, 1981, pp. 95–110

  9. Huxley, M.N.: Exponential sums and lattice points. III. Proc. London Math. Soc. (3) 87(3), 591–609 (2003)

  10. Jakobson D.: Quantum limits on flat tori. Ann. of Math. (2) 145, 235–266 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  11. Keating J.P., Marklof J., Winn B.: Localized eigenfunctions in S̆eba billiards. J. Math. Phys. 51(6), 062101 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  12. Kerckhoff, S., Masurm, H., Smillie, J.: Ergodicity of billiard flows and quadratic differentials. Ann. of Math. (2) 124(2), 293–311 (1986)

    Google Scholar 

  13. de L. Kronig R., Penney W.G.: Quantum Mechanics of Electrons in Crystal Lattices. Proce. Royal Society of London. Series A 130(814), 499–513 (1931)

    Article  ADS  Google Scholar 

  14. Landau E.: Über die Einteilung der positiven ganzen Zahlen in vier Klassen nach der Mindeszahl der zu ihrer additiven Zusammensetzung erforderlichen Quadrate. Arch. Math. Phys. 13, 305–312 (1908)

    MATH  Google Scholar 

  15. Marklof J., Rudnick Z.: Almost all eigenfunctions of a rational polygon are uniformly distributed. J. Spectral Th. 2, 107–113 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Oravecz F., Rudnick Z., Wigman I.: The Leray measure of nodal sets for random eigenfunctions on the torus. Ann. l’Institut Fourier 58(1), 299–335 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Rahav S., Fishman S.: Spectral statistics of rectangular billiards with localized perturbations. Nonlinearity 15(5), 1541–1594 (2002)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  18. Reed, M., Simon, B.: Methods of Modern Mathematical Physics, Vol. 2: Fourier Analysis and Self- Adjointness. London: Academic Press, 1975

  19. S̆eba P.: Wave Chaos in Singular Quantum Billiard. Phys. Rev. Let. 64(16), 1855–1858 (1990)

    Article  MathSciNet  ADS  Google Scholar 

  20. Shigehara T.: Conditions for the appearance of wave chaos in quantum singular systems with a pointlike scatterer. Phys. Rev. E 50, 4357–4370 (1994)

    Article  ADS  Google Scholar 

  21. Snirel’man, A.: Ergodic properties of eigenfunctions. Usp. Mat. Nauk 29(6), (180), 181–182 (1974)

    Google Scholar 

  22. Corput van der J.G: Neue zahlentheoretische Abschätzungen. Math. Ann. 89(3–4), 215–254 (1923)

    Article  MathSciNet  Google Scholar 

  23. Zelditch S.: Uniform distribution of eigenfunctions on compact hyperbolic surfaces. Duke Math. J. 55(4), 919–941 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  24. Zelditch S., Zworski M.: Ergodicity of eigenfunctions for ergodic billiards. Commun. Math. Phys. 175, 673–682 (1996)

    Article  MathSciNet  ADS  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, Tel Aviv, 69978, Israel

    Zeév Rudnick & Henrik Ueberschär

Authors
  1. Zeév Rudnick
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Henrik Ueberschär
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zeév Rudnick.

Additional information

Communicated by S. Zelditch

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rudnick, Z., Ueberschär, H. Statistics of Wave Functions for a Point Scatterer on the Torus. Commun. Math. Phys. 316, 763–782 (2012). https://doi.org/10.1007/s00220-012-1556-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-012-1556-2

Keywords

Search

Navigation