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Random waves and more: Eigenfunctions in chaotic and mixed systems

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Abstract.

The structure of wavefunctions strongly depends on the underlying classical dynamics. We illustrate this with several numerical examples and relate them to conjectures and mathematical results. In particular we focus on the random wave model and its implications for chaotic systems and its extension to systems with a mixed phase space.

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References

  • S. Tabachnikov, Billiards , Panoramas et Synthéses 1, Société Mathématique de France, Paris (1995)

  • Ya.G. Sinai, Sov. Math. Dokl. 4, 1818 (1963)

    Google Scholar 

  • Ya.G. Sinai, Russ. Math. Surv. 25, 137 (1970)

    Google Scholar 

  • L.A. Bunimovich, Funct. Anal. Appl. 8, 254 (1974)

    Google Scholar 

  • L.A. Bunimovich, Commun. Math. Phys. 65, 295 (1979)

    Google Scholar 

  • M. Wojtkowski, Commun. Math. Phys. 105, 391 (1986)

    Google Scholar 

  • D. Szász, Commun. Math. Phys. 145, 595 (1992)

    Google Scholar 

  • R. Markarian, Nonlinearity 6, 819 (1993)

  • H.-J. Stöckmann, Quantum chaos (Cambridge University Press, Cambridge, 1999)

  • A. Bäcker, Numerical aspects of eigenvalues and eigenfunctions of chaotic quantum systems , in: The Mathematical Aspects of Quantum Chaos I, edited by M. Degli Esposti S. Graffi (Springer) Lecture Notes Phys. 618, 91 (2003) and references therein

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

    Google Scholar 

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

    Google Scholar 

  • M.V. Berry, J. Phys. A 10 2083 (1977)

  • A. Voros, Semi-classical ergodicity of quantum eigenstates in the Wigner representation , in: Stochastic Behavior in Classical and Quantum Hamiltonian Systems CasFor79, 326

  • M.V. Berry, Semiclassical mechanics of regular and irregular motion , in: Comportement Chaotique des Systèmes Déterministes—Chaotic Behaviour of Deterministic Systems , edited by G. Iooss, R.H.G. Hellemann, R. Stora (North-Holland, Amsterdam, 1983), p. 171

  • A.I. Shnirelman, Usp. Math. Nauk 29, 181 (1974)

    Google Scholar 

  • S. Zelditch, Duke. Math. J. 55, 919 (1987)

    Google Scholar 

  • Y. Colin de Verdière, Commun. Math. Phys. 102, 497 (1985)

  • B. Helffer, A. Martinez, D. Robert, Commun. Math. Phys. 109, 313 (1987)

    Google Scholar 

  • P. Gérard, E. Leichtnam, Duke Math. J. 71, 559 (1993)

    Google Scholar 

  • S. Zelditch, M. Zworski, Commun. Math. Phys. 175, 673 (1996)

    Google Scholar 

  • A. Bäcker, R. Schubert, P. Stifter, Phys. Rev. E 57, 5425 (1998); erratum ibid. 58, 5192 (1998)

  • E.F.F. Chladni, Die Akustik, mit 12 Kupfertafeln, Breitkopf u. Härtel, Leipzig (1802)

  • S.W. McDonald, A.N. Kaufmann, Phys. Rev. A 37, 3067 (1988)

    Google Scholar 

  • P.W. O'Connor, E.J. Heller, Phys. Rev. Lett. 61, 2288 (1988)

    Google Scholar 

  • R. Aurich, F. Steiner, Physica D 64, 185 (1993)

    Google Scholar 

  • B. Li, M. Robnik, J. Phys. A 27, 5509 (1994)

    Google Scholar 

  • A. Bäcker, R. Schubert, J. Phys. A 35, 539 (2002)

    Google Scholar 

  • S. Hortikar, M. Srednicki, Phys. Rev. Lett. 80, 1646 (1998)

    Google Scholar 

  • B. Eckhardt, U. Dörr, U. Kuhl, H.-J. Stöckmann, Europhys. Lett. 46, 134 (1999)

    Google Scholar 

  • E.J. Heller, Phys. Rev. Lett. 53, 1515 (1984)

    Google Scholar 

  • A.N. Kolmogorov, Dokl. Akad. Nauk. SSSR 98, 527 (1954); English translation in CasFor79, 51

  • G. Casati, J. Ford (Eds.) Stochastic Behavior in Classical and Quantum Hamiltonian Systems , No. 93 in Lecture Notes in Physics (Springer-Verlag, Berlin, 1979)

  • V.I. Arnold, Russ. Math. Surv. 18, 9 (1963)

    Google Scholar 

  • J. Moser, Nachr. Akad. Wiss. Göttingen 1, 1 (1962)

    Google Scholar 

  • M. Robnik, J. Phys. A 16, 3971 (1983)

    Google Scholar 

  • H.R. Dullin, A. Bäcker, Nonlinearity 14, 1673 (2001)

    Google Scholar 

  • A. Bäcker, S. Fürstberger, R. Schubert, Phys. Rev. E 70, 036204 (2004)

    Google Scholar 

  • L. Hufnagel, R. Ketzmerick, M.-F. Otto, H. Schanz, Phys. Rev. Lett. 89, 154101 (2002)

    Google Scholar 

  • A. Bäcker, R. Ketzmerick, A. Monastra, Phys. Rev. Lett. 94, 054102 (2005)

    Google Scholar 

  • R. Ketzmerick, L. Hufnagel, F. Steinbach, M. Weiss, Phys. Rev. Lett. 85, 1214 (2000)

    Google Scholar 

  • A. Bäcker, R. Schubert, J. Phys. A 35, 527 (2002)

    Google Scholar 

  • M.V. Berry, H. Ishio, J. Phys. A 35, 5961 (2002)

    Google Scholar 

  • W.E. Bies, N. Lepore, E.J. Heller, J. Phys. A 36, 1605 (2003)

    Google Scholar 

  • J.D. Urbina, K. Richter, J. Phys. A 36, L495 (2003)

  • J.D. Urbina, K. Richter, Eur. Phys. J. Special Topics 145, 253 (2007)

    Google Scholar 

  • A. Bäcker, R. Ketzmerick, A. Monastra, Universality in the flooding of regular islands by chaotic states ArXiv nlin.CD/701032 (2007)

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Bäcker, A. Random waves and more: Eigenfunctions in chaotic and mixed systems. Eur. Phys. J. Spec. Top. 145, 161–169 (2007). https://doi.org/10.1140/epjst/e2007-00153-4

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