Bethe-Goldstone Equations in Scattering Theory

Abstract

The use of a continuum Bethe-Goldstone equation, recently proposed by Mittleman, to describe electron scattering by an alkali atom, is generalized by introducing the concept of continuum Bethe-Goldstone equations of successively higher order. In analogy to a method recently used for calculating mean-value properties of atomic stationary states, this makes possible the computation of net increments of a scattering amplitude or phase shift in successively higher orders, defined so that the sum of all net increments to order $N$ (for an $N$-particle system) is the exact amplitude or phase shift. Variational equations that might be used to solve a continuum Bethe-Goldstone equation of order $n$ are derived. Solution of a system of inhomogeneous linear equations is combined with integration of an integro-differential equation similar to a continuum Hartree-Fock equation. The formalism should be applicable to elastic scattering of an external particle by any many-fermion system.