Quantum Signatures of Chaos

  • Fritz Haake
Part of the NATO ASI Series book series (NSSB, volume 254)


The distinction between level clustering and level repulsion is one of the quantum analogues of the classical distinction between globally regular and predominantly chaotic motion (see Figs. 1, 2, 3). In order to reveal level repulsion under conditions of global classical chaos special care may be necessary: (i) subspectra referring to different values of the quantum numbers related to symmetries must be dealt with separately and (ii) for systems with quantum localization only levels whose wavefunctions have overlapping support must be admitted. A “level” may either be an energy eigenvalue E in the case of autonomous systems or, for periodically driven systems, a quasi-energy φ, i.e. an eigenphase of the unitary Floquet operator transporting the wavevector from period to period.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Statistical Theory of Spectra, C.E. Porter, ed., Academic Press, New York (1965)Google Scholar
  2. [2]
    M.L. Mehta, Random Matrices and the Statistical Theory of Spectra, Academic Press, New York (1965)Google Scholar
  3. [3]
    F. Haake, ‘Quantum Signatures of Chaos’, Springer-Verlag, Berlin, to appear (1990)Google Scholar
  4. [4]
    E.P. Wigner, Group Theory and its Applications to the Quantum Mechanics of Atomic Spectra, Academic Press, New York (1959)Google Scholar
  5. [5]
    M.V. Berry, in: Chaotic Behavior of Deterministic Systems, G. Iooss, R.H.G. Hellemann, and R. Stora, eds., Les Houches Session XXXVI, 1981, North Holland, Amsterdam (1983)Google Scholar
  6. [6]
    R. Scharf, B. Dietz, M. Kris, F. Haake, and M.V. Berry, Europhys. Lett. 5, 383 (1988)ADSCrossRefGoogle Scholar
  7. [7]
    P. Pechukas, Phys. Rev. Lett. 51, 943 (1983)ADSCrossRefMathSciNetGoogle Scholar
  8. [8]
    F. Haake, M. Küs, and R. Scharf, Lecture Notes in Phys. 282, F. Ehlotzky, ed., Springer, Berlin (1987)Google Scholar
  9. [9]
    M. Küs, to be publishedGoogle Scholar
  10. [10]
    F. Calogero and C. Marchioro, J. Math. Phys. 15, 1425 (1974)ADSCrossRefMathSciNetGoogle Scholar
  11. [11]
    B. Sutherland, Phys. Rev. A5, 1372 (1972)ADSCrossRefGoogle Scholar
  12. [12]
    J. Moser, Adv. Math. 16, 1 (1975)ADSCrossRefGoogle Scholar
  13. [13]
    M. Küs, Europhys. Lett. 5, 1 (1988)ADSCrossRefGoogle Scholar
  14. [14]
    P.D. Lax, Comm. Pure Appl. Math. 21, 467 (1968)CrossRefzbMATHMathSciNetGoogle Scholar
  15. [15]
    T. Yukawa, Phys. Rev. Lett. 54, 1883 (1985); Phys. Lett. 116 A, 227 (1986)ADSCrossRefGoogle Scholar
  16. [16]
    B. Dietz and F. Haake, Europhys. Lett. 9, 1 (1989); Z. Phys., to be publishedADSCrossRefGoogle Scholar
  17. [17]
    F. Llaake and G. Lenz, to be publishedGoogle Scholar
  18. [18]
    G. Lenz and F. Haake, to be publishedGoogle Scholar
  19. [19]
    H. Risken, The Fokker Planck Equation, Springer, Berlin (1984)CrossRefzbMATHGoogle Scholar
  20. [20]
    F. Dyson, J. Math. Phys. 3, 140 (1982)ADSCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • Fritz Haake
    • 1
  1. 1.Fachbereich PhysikUniversität-Gesamthochschule EssenEssen 1F. R. Germany

Personalised recommendations