Foundations of Physics

, Volume 31, Issue 3, pp 447–474 | Cite as

Periodic Orbit Quantization: How to make Semiclassical Trace Formulae Convergent

  • Jörg Main
  • Günter Wunner
Article

Abstract

Periodic orbit quantization requires an analytic continuation of non-convergent semiclassical trace formulae. We propose two different methods for semiclassical quantization. The first method is based upon the harmonic inversion of semiclassical recurrence functions. A band-limited periodic orbit signal is obtained by analytical frequency windowing of the periodic orbit sum. The frequencies of the periodic orbit signal are the semiclassical eigenvalues, and are determined by either linear predictor, Padé approximant, or signal diagonalization. The second method is based upon the direct application of the Padé approximant to the periodic orbit sum. The Padé approximant allows the resummation of the, typically exponentially, divergent periodic orbit terms. Both techniques do not depend on the existence of a symbolic dynamics, and can be applied to bound as well as to open systems. Numerical results are presented for two different systems with chaotic and regular classical dynamics, viz. the three-disk scattering system and the circle billiard.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

REFERENCES

  1. 1.
    M. C. Gutzwiller, J. Math. Phys. 8, 1979-2000 (1967); 12, 343-358 (1971).Google Scholar
  2. 2.
    M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer, New York, 1990).Google Scholar
  3. 3.
    M. V. Berry and M. Tabor, Proc. R. Soc. London A 349, 101-123 (1976); J. Phys. A 10, 371 (1977).Google Scholar
  4. 4.
    P. Cvitanović P and B. Eckhardt, Phys. Rev. Lett. 63, 823-826 (1989).Google Scholar
  5. 5.
    M. V. Berry and J. P. Keating, J. Phys. A 23, 4839-4849 (1990); Proc. R. Soc. London A 437, 151-173 (1992).Google Scholar
  6. 6.
    E. B. Bogomolny, Chaos 2, 5-13 (1992); Nonlinearity 5, 805-866 (1992).Google Scholar
  7. 7.
    R. Aurich, C. Matthies, M. Sieber, and F. Steiner, Phys. Rev. Lett. 68, 1629-1632 (1992).Google Scholar
  8. 8.
    J. Main, V. A. Mandelshtam, and H. S. Taylor, Phys. Rev. Lett. 79, 825-828 (1997).Google Scholar
  9. 9.
    J. Main, V. A. Mandelshtam, G. Wunner, and H. S. Taylor, Nonlinearity 11, 1015-1035 (1998).Google Scholar
  10. 10.
    J. Main and G. Wunner, Phys. Rev. A 59, R2548-R2551 (1999).Google Scholar
  11. 11.
    J. Main and G. Wunner, Phys. Rev. Lett. 82, 3038-3041 (1999).Google Scholar
  12. 12.
    J. Main, Phys. Rep. 316, 233-338 (1999).Google Scholar
  13. 13.
    J. Main, P. A. Dando, Dž. Belkić, and H. S. Taylor, J. Phys. A 33, 1247-1263 (2000).Google Scholar
  14. 14.
    M. R. Wall and D. Neuhauser, J. Chem. Phys. 102, 8011-8022 (1995).Google Scholar
  15. 15.
    V. A. Mandelshtam and H. S. Taylor, Phys. Rev. Lett. 78, 3274-3277 (1997).Google Scholar
  16. 16.
    V. A. Mandelshtam and H. S. Taylor, J. Chem. Phys. 107, 6756-6769 (1997); 109, 4128 (1998) (erratum).Google Scholar
  17. 17.
    J. Main, P. A. Dando, Dzž. Belkić, and H. S. Taylor, Europhys. Lett. 48, 250-256 (1999).Google Scholar
  18. 18.
    G. A. Baker, Essentials of Padé Approximants (Academic, New York, 1975).Google Scholar
  19. 19.
    J. Čížek and E. R. Vrscay, Int. J. Quantum Chem. 21, 27 (1982); H. J. Silverstone, B. G. Adams, J. C8 @ z ek, and P. Otto, Phys. Rev. Lett. 43, 1498 (1979).Google Scholar
  20. 20.
    Dž. Belkić, J. Phys. A 22, 3003-3010 (1989).Google Scholar
  21. 21.
    B. Eckhardt and G. Russberg, Phys. Rev. E 47, 1578-1588 (1993).Google Scholar
  22. 22.
    Dž. Belkić, P. A. Dando, J. Main, and H. S. Taylor, J. Chem. Phys. 113, 6542-6556 (2000).Google Scholar
  23. 23.
    S. D. Silverstein and M. D. Zoltowski, Digital Signal Processing 1, 161-175 (1991).Google Scholar
  24. 24.
    Dž. Belkić, P. A. Dando, H. S. Taylor, and J. Main, Chem. Phys. Lett. 315, 135-139 (1999); Dz. Belkic, P. A. Dando, J. Main, H. S. Taylor, and S. K. Shin, J. Phys. Chem. A 104, 11677-11684 (2000); ibid. 105, 514 (2001) (erratum).Google Scholar
  25. 25.
    J. Main, V. A. Mandelshtam, and H. S. Taylor, Phys. Rev. Lett. 78, 4351-4354 (1997).Google Scholar
  26. 26.
    B. Grémaud and D. Delande, Phys. Rev. A 61, 032504 (2000).Google Scholar
  27. 27.
    Baron Gaspard Riche de Prony, Essai expérimental et analytique: sur les lois de la dilatabilité de fluides élastique et sur celles de la force expansive de la vapeur de l'alkool, diffrentes températures, Journal de l'E cole Polytechnique, Vol. 1, cahier 22, 22-76 (1795).Google Scholar
  28. 28.
    W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes, 2nd ed. (Cambridge University Press, Cambridge, 1992).Google Scholar
  29. 29.
    B. Eckhardt, G. Russberg, P. Cvitanović, P. E. Rosenqvist, and P. Scherer, in Quantum Chaos between Order and Disorder, G. Casati and B. V. Chirikov, eds. (Cambridge University Press, Cambridge, 1995), pp.405-433.Google Scholar
  30. 30.
    A. Wirzba, Phys. Rep. 309, 1-116 (1999).Google Scholar
  31. 31.
    I. M. Longman, Int. J. Comp. Math. B 3, 53 (1971).Google Scholar
  32. 32.
    P. Wynn, Mathematical Tables and Other Aids for Computations 10, 91 (1956); SIAM J. Numer. Anal. 3, 91 (1966); E. J. Weniger, Comp. Phys. Rep. 10, 189-371 (1989).Google Scholar
  33. 33.
    D. Shanks, J. Math. Phys. 34, 1 (1955).Google Scholar
  34. 34.
    J. Main, K. Weibert, and G. Wunner, Phys. Rev. E 58, 4436-4439 (1998).Google Scholar
  35. 35.
    J. Main and G. Wunner, Phys. Rev. E 60, 1630-1638 (1999); J. Main, K. Weibert, V. A. Mandelshtam, and G. Wunner, ibid, 1639-1642 (1999).Google Scholar
  36. 36.
    K. Weibert, J. Main, and G. Wunner, Eur. Phys. J. D 12, 381-401 (2000).Google Scholar
  37. 37.
    P. Cvitanović and B. Eckhardt, Nonlinearity 6, 277-311 (1993).Google Scholar
  38. 38.
    A. Voros, J. Phys. A 21, 685-692 (1988).Google Scholar
  39. 39.
    P. Cvitanović, P. E. Rosenqvist, G. Vattay, and H. H. Rugh, Chaos 3, 619-636 (1993).Google Scholar
  40. 40.
    P. Cvitanović and G. Vattay, Phys. Rev. Lett. 71, 4138-4141 (1993).Google Scholar
  41. 41.
    R. Balian and C. Bloch, Ann. Phys. 69, 76-160 (1972).Google Scholar
  42. 42.
    S. M. Reimann, M. Brack, A. G. Magner, J. Blaschke, and M. V. N. Murthy, Phys. Rev. A 53, 39-48 (1996).Google Scholar
  43. 43.
    I. C. Percival, Adv. Chem. Phys. 36, 1-61 (1977).Google Scholar
  44. 44.
    S. Hortikar and M. Srednicki, Phys. Rev. E 61, R2180-R2183 (2000).Google Scholar

Copyright information

© Plenum Publishing Corporation 2001

Authors and Affiliations

  • Jörg Main
    • 1
  • Günter Wunner
    • 1
  1. 1.Institut für Theoretische Physik 1Universität StuttgartStuttgartGermany

Personalised recommendations