N-body quantum scattering theory in two Hilbert spaces. V. Computation strategy

Abstract

In this paper the development of our previously published theory of approximations for the Chandler–Gibson (CG) equations is continued. In particular, our approximation theory is rigorously brought to the point where N-particle scattering calculations can begin. This is accomplished by mapping the CG operator equations into a function equation form, where the unknowns belong to a new (third!) computational Hilbert space ℒ. This mapping is facilitated by rescaling the Jacobi momentum variables for the relative free motion of the asymptotic clusters so that surfaces of constant kinetic energy are hyperspheres. The input terms to the resulting equations are expanded in a basis on the surface of the kinetic energy hypersphere. This leads to a system of infinitely many coupled one-dimensional integral equations with the kinetic energy as the continuous variable. The half-on-shell variant of these equations is then transformed to a K-matrix form. Our approximations result from truncating this system to a finite number of equations, which is equivalent to using a finite basis approximation of the original input terms. The basis sets could be hyperspherical harmonics, but the use of hyperspherical spline functions is also proposed. Our method generalizes the well-known method of partial waves for channels with two clusters, and it accommodates breakup channels in a straightforward way.