Poincare's Theorem and Subdynamics for Time-Dependent Systems.

Large dynamical systems with explicit time-dependence are studied. For these systems, there exist resonances between internal frequencies and/or between internal and external frequencies. As in the time-independent case, the usual canonical or unitary perturbation theory leads to divergences due to the resonances. This is a generalization of Poincare's theorem on non-integrability and extends the notion of large Poincare systems, i.e. systems with a continuous spectrum and a continuous set of resonances. Using the natural time ordering and a new rule of integration, we can construct time-dependent projection operators which decompose the equations of motion and are analytical in the coupling conatant. These projectors are non-Hermitian and provide a description with broken time symmetry. As an application, we study the modification of the well-known three stages of the decay process from a time-dependent perturbation. We also develop a time-independent formalism to treat these time-dependent systems by introducing an auxiliary variable and extending the phase space. In this way, the Hamiltonian becomes time-independent and the original formulation of the time-independent subdynamics can be applied. The results obtained by these two approaches are completely the same. Based on the recent development in the second quantization for the complex spectral theory, we investigate the non-Markovian effects by studying an exactly soluble matter-field interaction model. The results show that the ratio of the non-Markovian to Markovian effects can be enhanced enormously by imposing a non-resonant external field. This mechanism renders an experimental investigation possible. The exact solution obtained in this model also justifies our perturbational scheme mentioned in above.