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The search for a quantum KAM theorem

  • Part III. Invited Papers Dedicated to Ilya Prigogine
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Abstract

The complex mechanisms by which nonlinear classical conservative systems undergo a transition from quasiperiodic to chaotic behavior are now fairly well understood. This transition is associated with a breakdown of quasi-constants of motion (KAM surfaces). There is growing evidence that similar mechanisms may govern the behavior of quantum systems. While K-type mixing behavior has not yet been found, there does appear to be a transition associated with the destruction of a quantum quasi-constant of motion (quantum KAM states) which changes qualitatively the spectrum of quantum systems.

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References

  1. I. Prigogine, A. Grecos, and Cl. George,Celest. Mech. 16, 489 (1977).

    Article  Google Scholar 

  2. T. Y. Petrosky,Phys. Rev. A 32, 3716 (1985).

    Google Scholar 

  3. A. N. Kolmogorov, inFoundations of Mechanics, R. Abraham, ed. (W. A. Benjamin, New York, 1967), Appendix D.

    Google Scholar 

  4. V. I. Arnold,Russ. Math. Surv. 18, 9 (1963);18, 85 (1963).

    Google Scholar 

  5. J. Moser,Nachr. Akad. Wiss. Göttingen II, Math. Phys. Kl. 1, 1 (1962).

    Google Scholar 

  6. E. Fermi:Collected Papers, Vol. II (University of Chicago Press, Chicago, 1965), p. 978.

    Google Scholar 

  7. M. Henon and C. Heiles,Astron. J. 69, 73 (1964).

    Google Scholar 

  8. C. H. Walker and J. Ford,Phys. Rev. 188, 416 (1969).

    Google Scholar 

  9. H. Goldstein,Classical Mechanics, 2nd edn. (Addison-Wesley, Reading, Massachusetts, 1980).

    Google Scholar 

  10. J. Moser,Stable and Random Motions in Dynamical Systems (Princeton University Press, Princeton, 1973).

    Google Scholar 

  11. H.-D. Meyer,J. Chem. Phys. 84, 3147 (1986).

    Google Scholar 

  12. B. Chirikov,Phys. Rep. 52, 263 (1979).

    Google Scholar 

  13. D. F. Escande and F. Doveil,J. Stat. Phys. 26, 257 (1981);Phys. Lett. A 83, 307 (1981);Phys. Scr. T2, 126 (1982).

    Google Scholar 

  14. D. F. Escande,Phys. Rep. 121, 165 (1985).

    Google Scholar 

  15. I. C. Percival, inNonlinear Dynamics and the Beam-Beam Interaction, M. Month and J. C. Herreda, eds. (Am. Inst. of Phys. Conf. Proc. No. 57, 302, 1979).

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

    Google Scholar 

  17. Y. G. Sinai,Russ. Math. Surv. 25, 137 (1970).

    Google Scholar 

  18. M. V. Berry, inChaotic Behavior in Deterministic Systems, G. Ioosset al., eds. (Les Houches Summer School Proceedings36) (North-Holland, Amsterdam, 1983).

    Google Scholar 

  19. J. Moser, inNeue Züricher Zeitung, May 14, 1975.

  20. R. B. White, inStatistical Physics and Chaos in Fusion Plasma, W. Horton and L. E. Reichl, eds. (Wiley, New York, 1984).

    Google Scholar 

  21. W. A. Lin, “External Field-Induced Chaos in Classical and Quantum Hamiltonian Systems,” Ph.D. Dissertation, University of Texas, December 1986.

  22. W. A. Lin and L. E. Reichl,Physica D 19, 145 (1986).

    Google Scholar 

  23. G. Duffing,Erzwungene Schwingungen bei Verunderlicher Eigenfrequenz (Braunschweig, Vieweg, 1918).

    Google Scholar 

  24. L. E. Reichl and W. M. Zheng,Phys. Rev. A 29, 2186 (1984);30, 1068 (1984).

    Google Scholar 

  25. A. Einstein,Verhandl. Dtsch. Phys. Ges. 19, 82 (1917).

    Google Scholar 

  26. E. Wigner, reproduced in C. E. Porter,Statistical Theories of Spectra: Fluctuations (Academic Press, New York, 1965), p. 200.

    Google Scholar 

  27. M. L. Mehta,Random Matrices (Academic Press, New York, 1967).

    Google Scholar 

  28. T. A. Brodyet al., Rev. Mod. Phys. 53, 385 (1981).

    Google Scholar 

  29. M. Berry,Ann. Phys. (N.Y.) 131, 163 (1981).

    Google Scholar 

  30. O. Bohigas, M. J. Giannoni, and C. Schmit,Phys. Rev. Lett. 52, 1 (1984).

    Google Scholar 

  31. T. Terasaka and T. Matsushita,Phys. Rev. A 32, 538 (1985).

    Google Scholar 

  32. M. V. Berry and M. Tabor,Proc. R. Soc. (London) A 356, 375 (1977).

    Google Scholar 

  33. G. Casati, B. V. Chirikov, and I. Guarneri,Phys. Rev. Lett. 54, 1350 (1985).

    Google Scholar 

  34. T. H. Seligman and J. J. M. Verbaarschot,Phys. Rev. Lett. 56, 2767 (1986).

    Google Scholar 

  35. W. A. Lin and L. E. Reichl, “Transition of Quasienergy Spectral Statistics Due to Nonlinear Quantum Resonance Overlap” (Preprint, University of Texas, December 1986).

  36. L. E. Reichl and W. A. Lin,Phys. Rev. A 33 (Rapid Commun.), 3598 (1986).

    Google Scholar 

  37. G. P. Berman and G. M. Zaslavskii,Phys. Lett. A 61, 295 (1977).

    Google Scholar 

  38. G. P. Berman, G. M. Zaslavskii, and R. Kolovsky,Phys. Lett. A 87, 152 (1982).

    Google Scholar 

  39. G. P. Berman and A. R. Kolovsky,Phys. Lett. A 95, 15 (1983).

    Google Scholar 

  40. G. Hose and H. S. Taylor,Phys. Rev. Lett. 51, 947 (1983).

    Google Scholar 

  41. G. Hose, H. S. Taylor, and A. Tip,J. Phys. A. Math. Gen. 17, 1203 (1984).

    Google Scholar 

  42. R. Ramaswamy,J. Chem. Phys. 80, 6194 (1984).

    Google Scholar 

  43. W. M. Zheng and L. E. Reichl, “Level Mixing in the Stretch Hydrogen Atom,”Phys. Rev. A 35 (Rapid Commun.), January 1987.

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Authors and Affiliations

  1. Center for Studies in Statistical Mechanics, The University of Texas, 78712, Austin, Texas

    L. E. Reichl & W. A. Lin

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  1. L. E. Reichl
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  2. W. A. Lin
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Reichl, L.E., Lin, W.A. The search for a quantum KAM theorem. Found Phys 17, 689–697 (1987). https://doi.org/10.1007/BF01889542

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  • DOI: https://doi.org/10.1007/BF01889542

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