Abstract:
The statistical properties of random analytic functions psi(z) are investigated as a phase-space model for eigenfunctions of fully chaotic systems. We generalize to the plane and to the hyperbolic plane a theorem concerning the equidistribution of the zeros of psi(z) previously demonstrated for a spherical phase space (SU(2) polynomials). For systems with time reversal symmetry, the number of real roots is computed for the three geometries. In the semiclassical regime, the local correlation functions are shown to be universal, independent of the system considered or the geometry of phase space. In particular, the autocorrelation function of psi is given by a Gaussian function. The connections between this model and the Gaussian random function hypothesis as well as the random matrix theory are discussed.

Abstract:
Chaotic deterministic dynamics of a particle can give rise to diffusive Brownian motion. In this paper, we compute analytically the diffusion coefficient for a particular two-dimensional stochastic layer induced by the kicked Harper map. The variations of the transport coefficient as a parameter is varied are analyzed in terms of the underlying classical trajectories with particular emphasis in the appearance and bifurcations of periodic orbits. When accelerator modes are present, anomalous diffusion of the L\'evy type can occur. The exponent characterizing the anomalous diffusion is computed numerically and analyzed as a function of the parameter.

Abstract:
Fully chaotic Hamiltonian systems possess an infinite number of classical solutions which are periodic, e.g. a trajectory ``p'' returns to its initial conditions after some fixed time tau_p. Our aim is to investigate the spectrum tau_1, tau_2, ... of periods of the periodic orbits. An explicit formula for the density rho(tau) = sum_p delta (tau - tau_p) is derived in terms of the eigenvalues of the classical evolution operator. The density is naturally decomposed into a smooth part plus an interferent sum over oscillatory terms. The frequencies of the oscillatory terms are given by the imaginary part of the complex eigenvalues (Ruelle--Pollicott resonances). For large periods, corrections to the well--known exponential growth of the smooth part of the density are obtained. An alternative formula for rho(tau) in terms of the zeros and poles of the Ruelle zeta function is also discussed. The results are illustrated with the geodesic motion in billiards of constant negative curvature. Connections with the statistical properties of the corresponding quantum eigenvalues, random matrix theory and discrete maps are also considered. In particular, a random matrix conjecture is proposed for the eigenvalues of the classical evolution operator of chaotic billiards.

Abstract:
Shell effects in atomic nuclei are a quantum mechanical manifestation of the single--particle motion of the nucleons. They are directly related to the structure and fluctuations of the single--particle spectrum. Our understanding of these fluctuations and of their connections with the regular or chaotic nature of the nucleonic motion has greatly increased in the last decades. In the first part of these lectures these advances, based on random matrix theories and semiclassical methods, are briefly reviewed. Their consequences on the thermodynamic properties of Fermi gases and, in particular, on the masses of atomic nuclei are then presented. The structure and importance of shell effects in the nuclear masses with regular and chaotic nucleonic motion are analyzed theoretically, and the results are compared to experimental data. We clearly display experimental evidence of both types of motion

Abstract:
It is well known that the joint probability density of the eigenvalues of Gaussian ensembles of random matrices may be interpreted as a Coulomb gas. We review these classical results for hermitian and complex random matrices, with special attention devoted to electrostatic analogies. We also discuss the joint probability density of the zeros of polynomials whose coefficients are complex Gaussian variables. This leads to a new two-dimensional solvable gas of interacting particles, with non-trivial interactions between particles.

Abstract:
The time evolution of a bounded quantum system is considered in the framework of the orthogonal, unitary and symplectic circular ensembles of random matrix theory. For an $N$ dimensional Hilbert space we prove that in the large $N$ limit the return amplitude to the initial state and the transition amplitude to any other state of Hilbert space are Gaussian distributed. We further compute the exact first and second moments of the distributions. The return and transition probabilities turn out to be non self-averaging quantities with a Poisson distribution. Departures from this universal behaviour are also discussed.

Abstract:
We investigate the statistical distribution of the zeros of Dirichlet $L$--functions both analytically and numerically. Using the Hardy--Littlewood conjecture about the distribution of prime numbers we show that the two--point correlation function of these zeros coincides with that for eigenvalues of the Gaussian unitary ensemble of random matrices, and that the distributions of zeros of different $L$--functions are statistically independent. Applications of these results to Epstein's zeta functions are shortly discussed.

Abstract:
We compute the dispersion laws of chaotic periodic systems using the semiclassical periodic orbit theory to approximate the trace of the powers of the evolution operator. Aside from the usual real trajectories, we also include complex orbits. These turn out to be fundamental for a proper description of the band structure since they incorporate conduction processes through tunneling mechanisms. The results obtained, illustrated with the kicked-Harper model, are in excellent agreement with numerical simulations, even in the extreme quantum regime.

Abstract:
We compute the stationary profiles of a coherent beam of Bose condensed atoms propagating through a guide. Special emphasis is put on the effect of an obstacle present on the trajectory of the beam. The obstacle considered (such as a bend in the guide, or a laser field perpendicular to the beam) results in a repulsive or an attractive potential acting on the condensate. Different behaviors are observed when varying the beam velocity (with respect to the speed of sound), the size of the obstacle (relative to the healing length) and the intensity and sign of the potential. The existence of bound states of the condensate is also considered.

Abstract:
Bifurcations of periodic orbits as an external parameter is varied are a characteristic feature of generic Hamiltonian systems. Meyer's classification of normal forms provides a powerful tool to understand the structure of phase space dynamics in their neighborhood. We provide a pedestrian presentation of this classical theory and extend it by including systematically the periodic orbits lying in the complex plane on each side of the bifurcation. This allows for a more coherent and unified treatment of contributions of periodic orbits in semiclassical expansions. The contribution of complex fixed points is find to be exponentially small only for a particular type of bifurcation (the extremal one). In all other cases complex orbits give rise to corrections in powers of $\hbar$ and, unlike the former one, their contribution is hidden in the ``shadow'' of a real periodic orbit.