Skip to main content
SpringerLink
Log in

Limited sensitivity to analyticity: a manifestation of quantum chaos

  • Published:
Foundations of Physics Aims and scope Submit manuscript

Abstract

Classical Hamiltonian systems generally exhibit an intricate mixture of regular and chaotic motions on all scales of phase space. As a nonintegrability parameter K (the “strength of chaos”) is gradually increased, the analyticity domains of functions describing regular-motion components [e.g., Kolmogorov-Arnol’d-Moser (KAM) tori] usually shrink and vanish at the onset of global chaos (breakup of all KAM tori). It is shown that these phenomena have quantum-dynamical analogs in simple but representative classes of model systems, the kicked rotors and the two-sided kicked rotors. Namely, as K is gradually increased, the analyticity domain ℛQE of the quantum-dynamical eigenstates decreases monotonically, and the width of ℛQE in the global-chaos regime vanishes in the semiclassical limit. These phenomena are presented as particular aspects of a more general scenario: As K is increased, ℛQE gradually becomes less sensitive to an increase in the analyticity domain of the system.

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. See, e.g., A. J. Dragt and J. M. Finn,J. Geoph. Res. 81, 2327 (1976), and references therein.

    Article  ADS  Google Scholar 

  2. M. V. Berry, inTopics in Nonlinear Dynamics, S. Jorna, ed.AIP Conf. Proc. 46 (AIP, New York, 1978).

    Google Scholar 

  3. B. V. Chirikov,Phys. Rep. 52, 263 (1979).

    Article  ADS  MathSciNet  Google Scholar 

  4. A. J. Lichtenberg and M. A. Lieberman,Regular and Chaotic Dynamics (Springer, New York, 1992).

    MATH  Google Scholar 

  5. R. S. MacKay and J. D. Meiss,Hamiltonian Dynamical Systems (Adam Hilger, Bristol, 1987).

    MATH  Google Scholar 

  6. C. F. F. Karney,Physica D 8, 360 (1983); J. D. Meiss and E. Ott,Physica D 20, 387 (1986), and references therein.

    Article  ADS  MathSciNet  Google Scholar 

  7. W. P. Reinhardt,J. Phys. Chem. 86, 2158 (1982); R. B. Shirts and W. P. Reinhardt,J. Chem. Phys. 77, 5204 (1982); N. Saito, H. Hirooka, J. Ford, F. Vivaldi, and G. H. Walker,Physica D 5 273 (1982); M. A. Sepulveda, R. Badii, and E. Pollak,Phys. Rev. Lett. 63, 1226 (1989); G. Stolovitzky and J. A. Hernando,Phys. Rev. A 43, 2774 (1991).

    Article  Google Scholar 

  8. I. Dana,Phys. Rev. Lett. 70, 2387 (1993).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. M. Gutzwiller,Chaos in Classical and Quantum Mechanics (Springer, Berlin, 1990), and references therein.

    MATH  Google Scholar 

  10. Quantum Chaos: between Order and Disorder, G. Casati and B. Chirikov, eds. (Cambridge University Press, Cambridge, 1995), and references therein.

    Google Scholar 

  11. D. K. Umberger and J. D. Farmer,Phys. Rev. Lett. 55, 661 (1985).

    Article  ADS  MathSciNet  Google Scholar 

  12. A. N. Kolmogorov,Dokl. Akad. Nauk SSSR 98, 527 (1954); V. I. Arnol’d,Izv. Akad. Nauk 25, 21 (1961); J. Moser,Nachr. Akad. Wiss. Göttingen Math. Phys. Kl. IIa, 1 (1962).

    MathSciNet  MATH  Google Scholar 

  13. R. S. MacKay, J. D. Meiss, and I. C. Percival,Physica D 13, 55 (1984).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. I. C. Percival, inNonlinear Dynamics and the Beam-Beam Interaction, M. Month and J. C. Herrera, eds.AIP Conf. Proc. 57, 302 (1979).

  15. J. M. Greene,J. Math. Phys. 20, 1183 (1979).

    Article  ADS  Google Scholar 

  16. I. Dana and S. Fishman,Physica D 17, 63 (1985); I. Dana, N. W. Murray, and I. C. Percival,Phys. Rev. Lett. 62, 233 (1989).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. R. S. MacKay and I. C. Percival,Commun. Math. Phys.,98, 469 (1985), and references therein.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. S. J. Shenker and L. P. Kadanoff,J. Stat. Phys. 27, 631 (1982).

    Article  MathSciNet  ADS  Google Scholar 

  19. S. N. Coopersmith and D. S. Fisher,Phys. Rev. B 28, 2566 (1983).

    Article  ADS  MathSciNet  Google Scholar 

  20. I. Dana and W. P. Reinhardt,Physica D 28, 115 (1987).

    Article  ADS  MathSciNet  Google Scholar 

  21. W. M. Zheng,Phys. Rev. A 33, R2850 (1986).

    Article  ADS  Google Scholar 

  22. E. Eisenberg and N. Shnerb,Phys. Rev. E. 49, R941 (1994).

    Article  ADS  Google Scholar 

  23. I. Dana, E. Eisenberg, and N. Shnerb,Phys. Rev. Lett. 74, 686 (1995); E. Eisenberg, N. Shnerb, and I. Dana, inCurrent Developments in Mathematics (International Press, Boston, 1996), pp. 81–87.

    Article  ADS  Google Scholar 

  24. I. Dana, E. Eisenberg, and N. Shnerb,Phys. Rev. E 54, 5948 (1996).

    Article  ADS  Google Scholar 

  25. F. M. Izrailev and D. L. Shepelyanskii,Theor. Math. Phys. 43, 553 (1980).

    Article  MathSciNet  Google Scholar 

  26. E. L. Ince,Ordinary Differential Equations (Dover, London, 1956); E. T. Whittaker and G. N. Watson,A Course of Modern Analysis, 4th ed. (University Press, Cambridge, 1952).

    Google Scholar 

  27. M. Abramowitz and I. A. Stegun,Handbook of Mathematical Functions (Dover, New York, 1972).

    MATH  Google Scholar 

  28. S. Fishman, D. R. Grempel, and R. E. Prange,Phys. Rev. Lett. 49, 509 (1982),Phys. Rev. A 29, 1639 (1984).

    Article  ADS  MathSciNet  Google Scholar 

  29. D. L. Shepelyansky,Phys. Rev. Lett. 56, 677 (1986);Physica D 28, 103 (1987).

    Article  ADS  MathSciNet  Google Scholar 

  30. R. Blümel, S. Fishman, M. Griniasti, and U. Smilansky, inQuantum Chaos and Statistical Nuclear Physics, Proceedings of the 2nd International Conference on Quantum Chaos, Curnevaca, Mexico, T. H. Seligmam and H. Nishioka, eds. (Lecture Notes in Physics, Vol. 263) (Springer, Heidelberg, 1986), p. 212.

    Google Scholar 

  31. M. Griniasty and S. Fishman,Phys. Rev. Lett. 60, 1334 (1988); N. Brenner and S. Fishman,Nonlinearity 4, 211 (1992).

    Article  ADS  Google Scholar 

  32. N. F. Mott and W. D. Twose,Adv. Phys. 10, 107 (1961); P. Lloyd,J. Phys. C: Solid State Phys. 2, 1717 (1969); K. Ishii,Prog. Theor. Phys. Suppl. 53, 77 (1973).

    Article  ADS  Google Scholar 

  33. I. Dana and E. Eisenberg, to be published.

Download references

Author information

Author notes

  1. Itzhack Dana

    Present address: Pearl Resnick Advanced Technology Institute, Bar-Ilan University, 52900, Ramat-Gan, Israel

Authors and Affiliations

  1. Department of Physics, Bar-Ilan University, 52900, Ramat-Gan, Israel

    Eli Eisenberg & Itzhack Dana

Authors
  1. Eli Eisenberg
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Itzhack Dana
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Eisenberg, E., Dana, I. Limited sensitivity to analyticity: a manifestation of quantum chaos. Found Phys 27, 153–170 (1997). https://doi.org/10.1007/BF02550447

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02550447

Keywords

Search

Navigation