N-body quantum scattering theory in two Hilbert spaces. IV. Approximate equations

Abstract

A rigorous mathematical theory of approximations is developed for the time-independent transition operators of N-body multichannel nonrelativistic quantum scattering theory. New basic dynamical equations are derived and shown to specify uniquely the approximate time-independent transition operators. These operator equations represent coupled integral equations with compact kernels, but it is not assumed that the equations that determine the exact transition amplitudes have compact kernels. Convergence of sequences of these approximate time-independent transition operators to the exact transition operator is established in appropriate limits. Stability of the basic dynamical equations is proved. Resolvent-type equations and their relation to the limiting absorption principle are investigated. The relation of this theory to the Petryshyn theory of A-proper operators and to the Feshbach unified theory of nuclear reactions is discussed.