One-Dimensional Three-Body Scattering Problem Used as a Testing Ground for theK-Matrix Method for Scattering Reactions of Complex Systems

The scattering reactions of three equal-mass particles constrained to move in a straight line and interacting with each other via zero-range potentials have been analyzed on the basis of the extended $R$-matrix theory. The simplicity of the model facilitates an exposition of the complexities that result from the existence of rearrangement channels and from the possibility for breakup into three-body channels. The conventional expressions for the $K$ matrix and the $T$ matrix are derived on a rigorous basis. A practical method for approximating the continuum of three-body breakup channels by a discrete set is used to carry out a distorted-wave Born approximation (DWBA) $K$-matrix calculation of the probabilities for transmission, knockout, and breakup when one particle is incident on a bound state of the other two. This method is found to give much better results than a DWBA $T$-matrix calculation.