next up previous
Next: Romanovski Up: Abstracts Previous: Prosen

Introduction to quantum chaos of generic Hamiltonian systems

Marko Robnik

Center for Applied Mathematics and Theoretical Physics,
University of Maribor, Maribor, Slovenia

We shall review the basic aspects of complete integrability and complete chaos (ergodicity) in classical Hamiltonian systems, as well as all the cases in between, the generic, mixed type systems, where KAM Theory is applicable, and shall illustrate it using the billiard model systems.

Then we shall proceed to the quantum chaos and its stationary properties, that is the structure and the morphology of the solutions of the underlying Schroedinger equation which in case of 2-dim billiards is just the 2-dim Helmholtz equation. We shall discuss the statistical properties of chaotic eigenfunctions, the statistical properties of the energy spectra, and show arguments and results in support of the so-called universality classes of spectral fluctuations, where in the fully chaotic case the Random Matrix Theory (RMT) is applicable.

First we discuss the universality classes of spectral fluctuations (GOE/GUE for ergodic systems, and Poissonian for integrable systems). We explain the problems in the calculation of the invariant (Liouville) measure of classically chaotic components, which has recently been studied by Robnik et al (1997) and by Prosen and Robnik (1998). Then we describe the Berry-Robnik (1984) picture, which is claimed to become exact in the strict semiclassical limit $\hbar\rightarrow 0$. However, at not sufficiently small values of $\hbar$ we see a crossover regime due to the localization properties of stationary quantum states where Brody-like behaviour with the fractional power law level repulsion is observed in the corresponding quantal energy spectra.

We shall mention the rich variety of applications in the domain of physics.

References
Aurich R, Bäcker A and Steiner F 1997 Int. J. Mod. Phys. 11 805
Berry M V 1983 in Chaotic Behaviour of Deterministic Systems eds. G Iooss, R H G Helleman and R Stora (Amsterdam: North-Holland) pp171-271
Berry M V 1991 in Chaos and Quantum Physics eds. M-J Giannoni, A Voros and J Zinn-Justin (Amsterdam: North-Holland) pp251-303
Berry M V and Robnik M 1984 J. Phys. A: Math. Gen. 17 2413
Bohigas O 1991 in Chaos and Quantum Physics eds. M-J Giannoni, A Voros and J Zinn-Justin (Amsterdam: North-Holland) pp87-199
Bohigas O, Giannoni M.-J. and Schmit C 1984 Phys. Rev. Lett. 25 1
Casati G and Chirikov B V 1994 in Quantum Chaos: Between Order and Disorder eds. G. Casati and B.V. Chirikov (Cambridge: Cambridge University Press)
Guhr T, Müller-Groeling A and Weidenmüller H A 1998, Phys.Rep. 299 189
Li Baowen and Robnik M 1994 J. Phys. A: Math. Gen. 27 5509
Li Baowen and Robnik M 1995a J. Phys. A: Math. gen. 28 2799
Li Baowen and Robnik M 1995b J. Phys. A: Math. gen. 28 4843
Prosen T and Robnik M 1993a J. Phys. A: Math. Gen. 26 L319
Prosen T and Robnik M 1993b J. Phys. A: Math. Gen. 26 1105
Prosen T and Robnik M 1993c J. Phys. A: Math. Gen. 26 2371
Prosen T and Robnik M 1993d J. Phys. A: Math. Gen. 26 L37
Prosen T and Robnik M 1994a J. Phys. A: Math. Gen. 27 L459
Prosen T and Robnik M 1994b J. Phys. A: Math. Gen. 27 8059
Robnik M and Prosen T 1997 J. Phys. A: Math. Gen. 30 8787
Robnik M 1984 J. Phys. A: Math. Gen. 17 1049
Robnik M 1988 in "Atomic Spectra and Collisions in External Fields", eds. K T Taylor, M H Nayfeh and C W Clark, (New York: Plenum) pp265-274
Robnik M 1998 Nonlinear Phenomena in Complex Systems 1 1
Veble G, Robnik M and Liu Junxian 2000 J. Phys. A: Math. Gen. 32 6423
Veble G, Kuhl U, Robnik M, H.-J. Stöckmann, Liu Junxian and Barth M 2000 Prog. Theor. Phys, Suppl. (Kyoto) 139 283
Veble G, Robnik M and Romanovski V 2002 J.Phys.A: Math.Gen. 35 4151


next up previous
Next: Romanovski Up: Abstracts Previous: Prosen