Recent results on time-dependent Hamiltonian oscillators

Time-dependent Hamilton systems are important in modeling the nondissipative interaction of the system with its environment. We review some recent results and present some new ones. In time-dependent, parametrically driven, one-dimensional linear oscillator, the complete analysis can be performed (in the sense explained below), also using the linear WKB method. In parametrically driven nonlinear oscillators extensive numerical studies have been performed, and the nonlinear WKB-like method can be applied for homogeneous power law potentials (which e.g. includes the quartic oscillator). The energy in time-dependent Hamilton systems is not conserved, and we are interested in its evolution in time, in particular the evolution of the microcanonical ensemble of initial conditions. In the ideal adiabatic limit (infinitely slow parametric driving) the energy changes according to the conservation of the adiabatic invariant, but has a Dirac delta distribution. However, in the general case the initial Dirac delta distribution of the energy spreads and we follow its evolution, especially in the two limiting cases, the slow variation close to the adiabatic regime, and the fastest possible change – a parametric kick, i.e. discontinuous jump (of a parameter), where some exact analytic results are obtained (the so-called PR property, and ABR property). For the linear oscillator the distribution of the energy is always, rigorously, the arcsine distribution, whose variance can in general be calculated by the linear WKB method, while in nonlinear systems there is no such universality. We calculate the Gibbs entropy for the ensembles of noninteracting nonlinear oscillator, which gives the right equipartition and thermostatic laws even for one degree of freedom.