Subdynamics and nonintegrable systems

Work on the application of Poincare's theorem to large classical or quantum systems with a continuous spectrum is continued. In situations where it is applicable, Poincare's theorem prevents the construction of a complete set of eigenprojectors which would be hermitian as well as analytic in the coupling constant. In contrast, the theory of subdynamics as developed by the Brussels group permits the construction of a unique set of projectors П(ν)(for t > 0), giving up the requirement of hermiticity which is replaced by “star-hermiticity”. The theory of subdynamics is presented in a new self-contained way, starting from the commutation relation П(ν) LH=LH П(ν), where LH is the Liouvillian. This presentation is far more direct, and avoids some of the lengthy discussions associated with previous presentations (based mainly on the resolvent of the Liouvillian). Subdynamics appears to be of interest from many points of view. It generalizes the concept of spectral representation while permitting to retain all the degrees of freedom present in the unperturbed Hamiltonian. In contrast, degrees of freedom are lost when going to the spectral representation (e.g. in the Friedrichs model). Subdynamics permits us to solve the initial value problem associated with the Liouville equation retaining the “non-Markovian” contributions which appear in the standard presentation. Finally, it introduces a classification of large dynamical systems, classical or quantum, into integrable and nonintegrable ones. It is therefore of direct interest for a number of basic problems which belong to the class of nonintegrable dynamical systems, such as the interaction of matter with light. The applications of this technique to these problems will be worked out in subsequent papers.