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.
Similar content being viewed by others
References
S. Tabachnikov, Billiards , Panoramas et Synthéses 1, Société Mathématique de France, Paris (1995)
Ya.G. Sinai, Sov. Math. Dokl. 4, 1818 (1963)
Ya.G. Sinai, Russ. Math. Surv. 25, 137 (1970)
L.A. Bunimovich, Funct. Anal. Appl. 8, 254 (1974)
L.A. Bunimovich, Commun. Math. Phys. 65, 295 (1979)
M. Wojtkowski, Commun. Math. Phys. 105, 391 (1986)
D. Szász, Commun. Math. Phys. 145, 595 (1992)
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)
M.V. Berry, M. Tabor, Proc. R. Soc. London Ser. A 356, 375 (1977)
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)
S. Zelditch, Duke. Math. J. 55, 919 (1987)
Y. Colin de Verdière, Commun. Math. Phys. 102, 497 (1985)
B. Helffer, A. Martinez, D. Robert, Commun. Math. Phys. 109, 313 (1987)
P. Gérard, E. Leichtnam, Duke Math. J. 71, 559 (1993)
S. Zelditch, M. Zworski, Commun. Math. Phys. 175, 673 (1996)
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)
P.W. O'Connor, E.J. Heller, Phys. Rev. Lett. 61, 2288 (1988)
R. Aurich, F. Steiner, Physica D 64, 185 (1993)
B. Li, M. Robnik, J. Phys. A 27, 5509 (1994)
A. Bäcker, R. Schubert, J. Phys. A 35, 539 (2002)
S. Hortikar, M. Srednicki, Phys. Rev. Lett. 80, 1646 (1998)
B. Eckhardt, U. Dörr, U. Kuhl, H.-J. Stöckmann, Europhys. Lett. 46, 134 (1999)
E.J. Heller, Phys. Rev. Lett. 53, 1515 (1984)
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)
J. Moser, Nachr. Akad. Wiss. Göttingen 1, 1 (1962)
M. Robnik, J. Phys. A 16, 3971 (1983)
H.R. Dullin, A. Bäcker, Nonlinearity 14, 1673 (2001)
A. Bäcker, S. Fürstberger, R. Schubert, Phys. Rev. E 70, 036204 (2004)
L. Hufnagel, R. Ketzmerick, M.-F. Otto, H. Schanz, Phys. Rev. Lett. 89, 154101 (2002)
A. Bäcker, R. Ketzmerick, A. Monastra, Phys. Rev. Lett. 94, 054102 (2005)
R. Ketzmerick, L. Hufnagel, F. Steinbach, M. Weiss, Phys. Rev. Lett. 85, 1214 (2000)
A. Bäcker, R. Schubert, J. Phys. A 35, 527 (2002)
M.V. Berry, H. Ishio, J. Phys. A 35, 5961 (2002)
W.E. Bies, N. Lepore, E.J. Heller, J. Phys. A 36, 1605 (2003)
J.D. Urbina, K. Richter, J. Phys. A 36, L495 (2003)
J.D. Urbina, K. Richter, Eur. Phys. J. Special Topics 145, 253 (2007)
A. Bäcker, R. Ketzmerick, A. Monastra, Universality in the flooding of regular islands by chaotic states ArXiv nlin.CD/701032 (2007)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Issue Date:
DOI: https://doi.org/10.1140/epjst/e2007-00153-4