Skip to main content
SpringerLink
Log in

Periodic Orbits and Semiclassical Form Factor in Barrier Billiards

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

Abstract

Using heuristic arguments based on the trace formulas, we analytically calculate the semiclassical two-point correlation form factor for a family of rectangular billiards with a barrier of height irrational with respect to the side of the billiard and located at any rational position p/q from the side. To do this, we first obtain the asymptotic density of lengths for each family of periodic orbits by a Siegel-Veech formula. The result obtained for these pseudo-integrable, non-Veech billiards is different but not far from the value of 1/2 expected for semi-Poisson statistics and from values of obtained previously in the case of Veech billiards.

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. Agam, O., Altshuler, B.L., Andreev, A.V.: Spectral Statistics from disordered to chaotic systems. Phys. Rev. Lett 75, 4389 (1995)

    Article  PubMed  Google Scholar 

  2. Andreev, A.V., Altshuler, B.L.: Spectral Statistics beyond Random Matrix Theory. Phys. Rev. Lett. 75, 902–905 (1995)

    Article  PubMed  Google Scholar 

  3. Balian, R., Bloch, C.: Distribution of eigenfrequencies for the wave equation in a finite domain: Eigenfrequency density oscillations. Ann. Phys. (N.Y.) 69, 76 (1972)

    Article  Google Scholar 

  4. Berry, M.V.: Semiclassical theory of spectral rigidity. Proc. Roy. Soc. A 400, 229 (1985)

    Google Scholar 

  5. Richens, P.J., Berry, M.V.: Pseudointegrable systems in classical and quantum mechanics. Physica D 2, 495 (1981)

    Google Scholar 

  6. Berry, M.V., Tabor, M.: Closed Orbits and the Regular Bound Spectrum. Proc. Roy. Soc. Lond. A 349, 101–123 (1976)

    Google Scholar 

  7. Berry, M.V., Tabor, M.: Level clustering in the regular spectrum. Proc. Roy. Soc. Lond. A 356, 375–94 (1977)

    Google Scholar 

  8. Bohigas, O., Giannoni, M.-J., Schmit, C.: Characterization of Chaotic Quantum Spectra and Universality of Level Fluctuation Laws. Phys. Rev. Lett. 52, 1 (1984)

    Article  Google Scholar 

  9. Bogomolny, E.B.: Action correlations in integrable systems. Nonlinearity 13, 947–972 (2000)

    Article  Google Scholar 

  10. Bogomolny, E., Gerland, U., Schmit, C.: Models of intermediate spectral statistics. Phys. Rev. E 59, R1315 (1999)

  11. Bogomolny, E., Pavloff, N., Schmit, C.: Diffractive corrections in the trace formula for polygonal billiards. Phys. Rev. E 61, 3689 (2000)

    Article  Google Scholar 

  12. Bogomolny, E., Giraud, O., Schmit, C.: Periodic orbits contribution to the 2-point correlation form factor for pseudo-integrable systems. Commun. Math. Phys. 222, 327–369 (2001)

    Article  Google Scholar 

  13. Casati, G., Prosen, T.: Mixing Property of Triangular Billiards. Phys. Rev. Lett. 83, 4729–4732 (1999)

    Article  Google Scholar 

  14. Eskin, A., Masur, H.: Asymptotic formulas on flat surfaces. Ergod. Theor. & Dyn. Sys. 21, 443–478 (2001)

    Google Scholar 

  15. Eskin, A., Masur, H., Schmoll, M.: Billiards in Rectangles with Barriers. Duke Math. J. 118, 427–463 (2003)

    Article  Google Scholar 

  16. Eskin, A., Masur, H., Zorich, A.: Moduli spaces of abelian differentials: the principal boundary, counting problems and the Siegel-Veech constants. Publications IHES 97, 61–179 (2003)

    Article  Google Scholar 

  17. Giraud, O., Marklof, J., O’Keefe, S.: Intermediate statistics in quantum maps. J. Phys. A: Math. Gen. 37(28), L303–L311 (2004)

    Google Scholar 

  18. Giraud, O.: Statistiques spectrales des systèmes difractifs. PhD Thesis, Université Paris XI (2002)

  19. Gutkin, E., Judge, C.: Affine Mappings of Translation Surfaces: Geometry and Arithmetic. Duke Math. J. 103, 191–213 (2000)

    Article  Google Scholar 

  20. Gutzwiller, M.C.: The semi-classical quantization of chaotic hamiltonian systems. In: Chaos and Quantum Mechanics, Giannoni, M.-J., Voros, A., Zinn-Justin, J., (eds), Les Houches Summer School Lectures LII, 1989, Amsterdam: North Holland, 1991, pp. 87

  21. Hannay, J.H., McCraw, . J.: Barrier billiards - a simple pseudo-integrable system. J. Phys. A: Math. Gen. 23, 887–900 (1990)

    Article  Google Scholar 

  22. Hannay, J.H., Ozorio de Almeida, A. M.: Periodic orbits and a correlation function for the semiclassical density of states. J. Phys. A: Math. Gen. 17, 3429–3440 (1984)

    Article  Google Scholar 

  23. Huard, J.G., Ou, Z.M., Spearman, B. K., Williams, K.S.: Elementary Evaluation of Certain Convolution Sums involving Divisor Functions. Number Theory for the Millennium II, Natick, MA: A. K. Peters, 2002, pp. 229–274

  24. Marklof, J.: Spectral Form Factors of Rectangle Billiards. Commun. Math. Phys. 199, 169–202 (1998)

    Article  Google Scholar 

  25. Masur, H.: The Growth Rate of Trajectories of a Quadratic Differential. Ergod. Theor. & Dyn. Sys. 10, 151–176 (1990)

    Google Scholar 

  26. Sieber, M.: Geometrical theory of diffraction and spectral statistics. J. Phys. A: Math. Gen. 32, 7679–7689 (1999)

    Article  Google Scholar 

  27. Veech, W.A.: Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards. Invent. Math. 97, 553–583 (1989)

    Article  Google Scholar 

  28. Vorobets, Y.B.: Planar structures and billiards in rational polygons: the Veech alternative. Russ. Math. Surv. 51(5), 779–817 (1996)

    Google Scholar 

  29. Wiersig, J.: Spectral Properties of Quantized Barrier Billiards. Phys. Rev. E 65, 4627 (2002)

    Google Scholar 

  30. Zemlyakov, A.B., Katok, A.N.: Topological Transitivity of Billiards in Polygons. Math. Notes 18, 760–764 (1976)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire de Physique théorique, UMR 5152 du CNRS, Université Paul Sabatier, 31062, Toulouse Cedex 04, France

    O. Giraud

Authors
  1. O. Giraud
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by P. Sarnak

Rights and permissions

Reprints and permissions

About this article

Cite this article

Giraud, O. Periodic Orbits and Semiclassical Form Factor in Barrier Billiards. Commun. Math. Phys. 260, 183–201 (2005). https://doi.org/10.1007/s00220-005-1412-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-005-1412-8

Keywords

Search

Navigation