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 . However, at not sufficiently small values of 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