Skip to main content
SpringerLink
Log in

Quantum Ergodicity for Point Scatterers on Arithmetic Tori

  • Published:
Geometric and Functional Analysis Aims and scope Submit manuscript

Abstract

We prove an analogue of Shnirelman, Zelditch and Colin de Verdiè- re’s quantum ergodicity Theorems in a case where there is no underlying classical ergodicity. The system we consider is the Laplacian with a delta potential on the square torus. There are two types of wave functions: old eigenfunctions of the Laplacian, which are not affected by the scatterer, and new eigenfunctions which have a logarithmic singularity at the position of the scatterer. We prove that a full density subsequence of the new eigenfunctions equidistribute in phase space. Our estimates are uniform with respect to the coupling parameter, in particular the equidistribution holds for both the weak and strong coupling quantizations of the point scatterer.

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. Berkolaiko G., Keating J.P., Winn B.: No quantum ergodicity for star graphs. Communications in Mathematical Physics 250, 259–285 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bohigas O., Giannoni M.J., Schmit C.: Characterization of chaotic quantum spectra and universality of level fluctuation laws. Physical Review Letters 52, 1–4 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bogomolny E., Gerland U., Schmit C.: Singular statistics. Physical Review E 63(3), 036206 (2001)

    Article  MathSciNet  Google Scholar 

  4. Colinde Verdière Y.: Ergodicité et fonctions propres du laplacien. Communications in Mathematical Physics 102(3), 497–502 (1985)

    Article  MathSciNet  Google Scholar 

  5. Fainsilber L., Kurlberg P., Wennberg B.: Lattice points on circles and discrete velocity models for the Boltzmann equation. SIAM Journal on Mathematical Analysis 37(6), 1903–1922 (2006) (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  6. Hecke E.: Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen. Mathematische Zeitschrift 6(1–2), 11–51 (1920)

    Article  MathSciNet  Google Scholar 

  7. Keating J.P., Marklof J., Winn B.: Localised eigenfunctions in S̆eba billiards. Journal of Mathematical Physics 51, 062101 (2010)

    Article  MathSciNet  Google Scholar 

  8. P. Kurlberg and H. Ueberschär. Localised semiclassical measures for point scatterers on diophantine tori. (2014) (in preparation)

  9. E. Landau. Handbuch der Lehre von der Verteilung der Primzahlen. 2 Bände, 2nd edn. Chelsea Publishing Co., New York (1953) (with an appendix by Paul T. Bateman)

  10. Rieger G.J.: Aufeinanderfolgende Zahlen als Summen von zwei Quadraten. Nederlandse Akademie van Wetenschappen Proceedings of the Series A 68 = Indagationes Mathematicae 27, 208–220 (1965)

    MathSciNet  Google Scholar 

  11. Rudnick Z., Ueberschär H.: Statistics of wave functions for a point scatterer on the torus. Communications in Mathematical Physics 316(3), 763–782 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Rudnick Z., Ueberschär H.: On the eigenvalue spacing distribution for a point scatterer on a flat torus. Annales Henri Poincaré 15(1), 1–27 (2014)

    Article  MATH  Google Scholar 

  13. Ruzhansky M., Turunen V.: Pseudo-Differential Operators and Symmetries: Background Analysis and Advanced Topips. Birkhäuser, Basel (2010)

    Book  Google Scholar 

  14. S̆eba P.: Wave chaos in a singular quantum billiard. Physical Review Letters 64, 1855–1858 (1990)

    Article  MathSciNet  Google Scholar 

  15. T. Shigehara. Conditions for the appearance of wave chaos in quantum singular systems with a pointlike scatterer. Physical Review E, (6)50 (1994)

  16. Ueberschär H.: Quantum Chaos for point scatterers on flat tori. Philosophical Transactions of the Royal Society A 372, 20120509 (2014)

    Article  Google Scholar 

  17. Wirsing E.: Das Asymptotische Verhalten von Summen über multiplikative Funktionen. Mathematische Annalen 143, 75–102 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  18. N. Yesha. Quantum ergodicity for a point scatterer on the three-dimensional torus. Annales Henri Poincaré (2014). doi:10.1007/s00023-014-0318-4

  19. Zelditch S.: Uniform distribution of eigenfunctions on compact hyperbolic surfaces. Duke Mathematical Journal 55, 919–941 (1987)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, KTH Royal Institute of Technology, 10044, Stockholm, Sweden

    Pär Kurlberg

  2. Institut de Physique Théorique, CEA Saclay, 91191, Gif-sur-Yvette Cedex, France

    Henrik Ueberschär

Authors
  1. Pär Kurlberg
    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 Henrik Ueberschär.

Additional information

P. K. was partially supported by grants from the Göran Gustafsson Foundation and the Swedish Research Council.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kurlberg, P., Ueberschär, H. Quantum Ergodicity for Point Scatterers on Arithmetic Tori. Geom. Funct. Anal. 24, 1565–1590 (2014). https://doi.org/10.1007/s00039-014-0275-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00039-014-0275-6

Keywords

Search

Navigation