Semi-classical behaviour of Schrödinger's dynamics: Revivals of wave packets on hyperbolic trajectory

The aim of this paper is to study the semi-classical behaviour of Schrödinger's dynamics for an one-dimensional quantum Hamiltonian with a classical hyperbolic trajectory. As in the regular case (elliptic trajectory), we prove, that for an initial wave packets localized in energy, the dynamics follows the classical motion during short time. This classical motion is periodic and the period T_hyp is order of |ln h|. And, for large time, a new period T_rev for the quantum dynamics appears: the initial wave packets form again at t=T_rev. Moreover, for the time t=p/qT_rev a fractional revivals phenomenon of the initial wave packets appears: there is a formation of a finite number of clones of the original wave packet.