Syllabus
Classical Hamiltonian systems. Conditions of integrability. Regularity of motion of integrable systems. Actions and angles, periodical and quasiperiodical trajectories, rational and irrational tori. Poincare surface of section
Perturbations of integrable systems. Convergency of perturbation series. Problem of small denominators. Sufficiently irrational tori. The Kolmogorov-Arnold-Moser theorem. Fate of rational tori. The Birkhoff fixed-point theorem. Stable and instable trajectories. Lyapounov exponents, SALI and GALI methods.
Correspondence between classical and quantum mechanics. Propagators as integrals over paths. Semiclassical quantization of classically chaotic systems. Level density as the Gutzwiller sum over the classical peridic orbits.
Fluctuations of energy levels of quantum systems. Basic fluctuation measures: distribution of nearest-neighbor spacings, rigidity, number variance. Random matrix ensembles. Level fluctuations in GUE and GOE (Gaussian unitary ensemble, Gaussian orthogonal ensemble). Wigner surmise. Brody distribution. Scale invariance of a quantum spectrum and 1/f noise. Peres lattices. Bohigas-Giannoni-Schmit conjecture and its validity.
Annotation
Introductory lectures on basic properties of regular and chaotic motion in classical hamiltonian autonomous systems, on the semiclassical quantization of classical chaotic systems and on the spectral properties of random matrix ensembles. Good knowledge of the basis of classical and quantum mechanics is required.