Time { Dependent Scattering Theoryof N { Body Quantum

We give a full and selfcontained account of the basic results in N{body scattering theory which emerged over the last ten years: The existence and completeness of scattering states for potentials decreasing like r ? , > p 3 ? 1. Our approach is a synthesis of earlier work and of new ideas. Global conditions on the potentials are imposed only to deene the dynamics. Asymptotic completeness is derived from the fact that the mean square diameter of the system diverges like t 2 as t ! 1 for any orbit t which is separated in energy from thresholds and eigenvalues (a generalized version of Mourre's theorem involving only the tails of the potentials at large distances). We introduce new propagation observables which considerably simplify the phase{space analysis. As a topic of general interest we describe a method of commutator expansions.