Scattering Theory for the Quantum Envelope of a Classical System

Classical dynamics, reformulated in terms of its quantum envelope is studied for the stationary states of the interacting system. The dynamical variable of "elapsed time" plays a crucial role in this study. It is shown that the perturbation series for the elapsed time can be summed in various simple cases even when standard perturbation series diverge. For the special class of systems where the interactions fall off sufficiently fast at infinity, one could define "in" and "out" states; and consequently the wave matrices and scattering matrices. The scattering phase shifts bear a simple relation to the time delay in scattering.