Numerical Study of a Model Three-Body System

Abstract

An investigation is made of the properties of a simple three-body system consisting of three particles moving in one dimension and interacting through d-function potentials. The exact equations of three-particle scattering theory for this system are reduced without approximation to a set of three coupled one-dimensional integral equations which are solved numerically for a wide range of different potential strengths and particle masses. For the special case of identical particles the numerical solutions are compared with the exact solutions found previously by the author. The method of solution for general values of the parameters, which is based on computing the eigenvalue trajectories of the kernel of the scattering equations, allows a. systematic search for three-body bound states. In the case of nuclear or atomic-like configurations, a unique symmetric bound state is found and its binding energy computed. For molecular configurations, where there are two identical heavy particles interacting by the exchange of a third light particle, several excited states of both positive and negative parity are found and a comparison is made of their binding energies with the predictions of the adiabatic approximation. A reaction matrix formulation of the exact equations is used to calculate the probabilities of elastic and rearrangement scattering below the threshold for breakup. When the particles are identical, there is no elastic or rearrangement scattering in the backward direction. However, for particles of different mass or potentials of unequal strength, all kinematically possible scattering processes occur and the scattering properties of the model are quite complex. In particular an interesting feature of the calculations is the appearance of cusps in the elastic cross sections at the rearrangement threshold.