Extension of scattering theory for finite times: Three-body scattering.

Abstract

Scattering theory is extended to the description of processes during collision. Indeed, for large wave packets we may consider situations where time t, while large with respect to characteristic frequencies, is still smaller than the duration of the collision ${\mathit{t}}_{\mathit{c}}$. This leads to an asymptotic theory which is different from the usual S-matrix approach. There appear intermediate states with scattering cross sections which result from secular effects for t${\mathit{t}}_{\mathit{c}}$, but differ from the values obtained for t\ensuremath{\gtrsim}${\mathit{t}}_{\mathit{c}}$ in the S-matrix theory. The consideration of such intermediate time scales is the normal procedure in many-body situations as realized, for example, in chemical reactions. Our method is closely connected to our work on the extension of quantum theory beyond the Hilbert space for large Poincar\'e systems. This theory applies to persistent scattering using delocalized density matrices (an example is plane waves). For this case, the S-matrix theory is not valid, while the results of our asymptotic approach remain valid for all times. Our theoretical predictions have been validated by numerical simulations. \textcopyright{} 1996 The American Physical Society.