MULTIPLE-SCATTERING ANALYSIS OF THE THREE-BODY SCATTERING PROBLEM

Several arguments are presented to show that it would be of great value to have a theoretical expression for the three-body scattering amplitude in terms of free-particle two-body scattering amplitudes for the pion-deuteron, kaon- deuteron, and nucleon-deuteron scattering problems. Theoretical investigations of these three-body scattering problems are reviewed briefly; detailed discussions of various methods of solving the two-body scattering problem and of previous treatments of the three-body scattering problem with fixed target particles are given. The problem of scattering of a scalar particle by a target composed of two scalar particles all of which interact via scalar potentials is reduced to the problem of the scattering of a particle by two potentials by the assumption that the target particles are infinitely massive compared to the incident particle. The assumption is made that the two potentials do not overlap. The Schroedinger equation is converted into a pair of coupled integral equations for the wave scattered by each well. The Green's function in each of these equations is just the total Green's function for the two-body scattering problem. The coupled integral equations are solved first for square well potentials and S- wave scattering; then for square well potentials and scattering of themore » first N partial waves; and, finally, for any two spherically symmetric wells with finite radii and scattering of the flrst N partial waves. A test of the separable potential approximation is made for S-wave scattering by two overlapping square wells. The effect of the recoil of the target particles is considered. The scattering amplitude for the general three-body problem is obtained as a multiple- scattering expansion in terms of bound-target twobody scattering amplitudes. These bound-target two-body amplitudes are reduced to free-particle two-body amplitudes for the special case in which there is no interaction between the target particles. The usual separable potential approximation is applied to the general three-body problem, but the multiple-scattering expansion of the total elastic scattering amplitude still contains unknown operators. A more general form of the separable potential approximation is applied to the general three- body problem, and the total elastic scattering amplitude is obtained in terms of a multiple-scattering expansion that contains the free-particle two-body amplitudes and certain average values of known operators. It is argued that this result includes some of the effects of the recoil and binding of the target particles.« less