New developments in N-body scattering theory III

In the framework of the new approach to the N-body problem developed in the first two papers of this series we carry out exact effective few-cluster reductions of N-body Faddeev-Yakubovskiî-type equations. In particular we derive a single effective one-vector variable Lippmann-Schwinger equation in correspondence to a dominant two-cluster partition, and effective two-vector variable three-body Karlsson-Zeiger equations in correspondence to a dominant three-cluster partition. The effective interactions appearing in these few-cluster models can be evaluated through a perturbative solution of Faddeev-Yakubovskiî-type auxiliary equations. In our chain-of-partition-labelled approach we introduce several kinds of elementary transition operators which can be simply related to the physical transition amplitudes. Among them a privileged role is played by proper left asymmetric elementary transition operators: they lead to few-cluster approximation schemes having exactly the same structure as the usual phenomenological models.