3-BODY PHASE-SHIFT IN ONE-DIMENSIONAL 2-SCATTERING(1)

For a model of three particles on a line, subject to attractive delta- function interactions, we consider the 2 + 1 phase shift. We do this f rom the point of view of the calculation of the S-matrix in a hypersph erical adiabatic basis (an adiabatic S-matrix), and for energies rangi ng from the (negative) energy of the two-body bound state to a total e nergy of zero. We derive analytical expansions and present numerical w ork, for different approximations, and compare with the exact results that we obtain from the work of McGuire, whose model we have borrowed. We show that the simplest adiabatic approximation gives results that are qualitatively wrong, but that better approximations yield, for mos t of our range, excellent agreement with the exact result. Understandi ng the threshold behaviour, however, requires a zero-energy three-body bound state, or resonance, previously unsuspected for this model. The methods developed for the case of the simplest adiabatic approximatio n also yield threshold and low-energy results applicable to the two-bo dy problem in two dimensions.