Integrability and Chaos in classical and quantum systems

In this field the integrability and the ergodic properties of classical and quantum systems, both with finite and infinite degrees of freedom, were studied.

In the papers [6, 7,9, 10,P1,A1], the integrability problem was considered from a geometrical point of view. In particular, it was shown that the existence of an infinite set of conservation laws, as well as the existence of an Hamiltonian structure, are consequences of the existence of an invariant tensor field on the phase manifold. This analysis was done for soliton equations like KdV, sine-Gordon etc. and for dissipative systems like Burgers equation. The integrability structure has been studied in details also in terms of Lax pairs. In [51], it was shown that among the three known cases of integrability of the Henon-Heiles system, two are connected by a canonical transformation. In the work [62], a method which allows to find non trivial Lax pairs for separable, multi-periodic, dynamical systems on two tori, was proposed. Among the physical problems which can be treated by this method there are: the dynamics of a charged particle in a Coulomb potential and in presence of an external field, the Henon-Heiles model and the Chaplygin top.

In [32,33], the Quantum Inverse Scattering Method (algebraic Bethe Ansatz) has been applied to a discrete version of the non linear Schrödinger equation and results compared with those obtained by methods based on the symmetry properties of the system [P8], . From this comparison emerged the possibility the express the energy spectrum by means of symmetric functions of the Bethe equation solutions. In [36], it was shown that this result is valid for all models which can be solved via Bethe Ansatz.

In the set of papers [23, 25-28, 34, P4,P9,A2], the ergodic properties of quantum systems have been studied. In particular in [23, 25], it was shown that the level distribution of the quantum discrete self-trapping (DST) system follows the Poisson or the Wigner statistics depending on whether the corresponding classical limit is integrable or completely chaotic. Moreover, the hypothesis of Berry and Robnick on the level distribution for systems with a mixed phase space has been verified. In [34], this analysis was extended to the case of systems which in the classical limit exhibit the phenomenon of Arnold diffusion, while in the papers [28,P4], the problem of quantum chaos for the DST equation has been considered in terms of the Wigner functions of the Hamiltonian eigenstates. In the study of the ergodic properties of quantum systems an important role is played by the unstable periodic orbits of the corresponding classical system. To this end, in [31], a method which allows to determine the unstable periodic orbits of one dimensional chaotic maps was introduced. This algorithm is based on suitable continuous dynamics whose stationary points are in correspondence with the unstable orbits of the map. This approach can be generalized to the case of maps with higher dimensions for which a quantum description can be given.

The problem of the phase transition from regular to chaotic behavior, which is observed in many physical systems, was also investigated, both from the analytical and numerical point of view. In particular in [13], the transition to chaos observed in the double sine-Gordon system was studied by means of the Menlikov method. In [21,22] the chaotic behavior of breather-kink (antikink) solutions of the sine-Gordon system was analyzed by using an appropriate finite-dimensional reduced system. It was shown, in [22], that the existence of a separatrix in the reduced phase space can be a possible source of chaos for the infinite-dimensional system. In the set of papers [16, 17,P2, P3], the transition chaos-order-chaos for the DST equation was investigated. This analysis revealed the possibility of having very sharp order to chaos transitions, as well as the existence, in the parameter space, of order windows in regions characterized by a chaotic behavior.

The theory of dynamical systems was applied to the analysis of experimental data. In particular, in the work [48], the seismic signals released from the area of "Campi Flegrei" (Naples, Italy) was studied by means of the correlation dimension technique for the reconstruction of the phase space from a time serie.

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