Quantum non-linear oscillations

Abstract

The theory of classical non-linear oscillations is a fascinating field of theoretical and experimental research (see the classic [207] [19] or the recent review [297]). In particular, the transition from a regular periodic behaviour into a quasi-periodic and subsequently into an unstable chaotic behaviour (under a variation of some parameters) is a complex but striking phenomenon. One could expect that the classical behaviour of trajectories should be relevant to quantum wave functions. The correspondence between classical periodic orbits of integrable systems and the spectrum of the quantum Hamiltonian has been shown in [174] [75] [9]. The role of unstable periodic trajectories in quantum mechanics has been studied by the group of Heller [212]. It has been shown (mainly through numerical simulations) that the classical instability leads to a concentration of the Wigner function along certain curves (scars).