Theory of cluster approximations

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Abstract
Calculations of the scattering matrix for nonrelativistic multiparticle collision processes are often based on a decomposition of the system into interacting clusters, with a particular form assumed for the effective interaction among the clusters. This approach is formalized here by the definition of an exact effective interaction operator for the general multichannel scattering problem. While other formal definitions can be given, based, e.g., on Faddeev equations or on Feshbach projection-operator techniques, the approach described here is particularly well suited for the introduction of variational approximations. The variational property of the approximate effective potential can be maintained even when the bound-state wave functions of the subsystems are known only approximately. Furthermore, for sufficiently low scattering energies each trial function which enters into the calculation can be systematically improved using a rigorous subsidiary minimum principle of the Rayleigh-Ritz type.