Toward a Rigorous Justification of the Three-Body Impact Parameter Approximation

The impact parameter (IP) approximation is a semiclassical model in quantum scattering theory wherein N large masses interact with one small mass. We study this model in one spatial dimension using the tools of time-dependent scattering theory, considering a system of two large-mass particles and one small-mass particle. We demonstrate that the model's predictive power becomes arbitrarily good as the masses of the two heavy particles are made larger by studying the S-matrix for a particular scattering channel. We also show that the IP wave functions can be made arbitrarily close to the full three-body solution, uniformly in time, provided one of the large masses is fixed in place, and that such a result probably will not hold if we allow all the masses to move.