Dispersion-Theoretic Impulse Approximation for Potential Scattering

Abstract

In the impulse approximation for the scattering of a particle by a bound system, the amplitude is a sum of integrals over two-body scattering amplitudes, off the energy shell, folded into bound-state wave functions. In the usual formulation, the nonphysical two-body amplitudes are replaced by physical amplitudes with no firm justification for this procedure. The dispersion-theoretic formulation presented here, for elastic scattering, removes this difficulty; for low values of $t$, the momentum transfer squared, the discontinuity across the cut in the $t$ plane can be expressed in terms of the absorptive part of the physical two-body amplitude and the asymptotic form of the bound-state wave function. Working with a nonrelativistic model, it is shown that the Cutkosky method for finding absorptive parts of Feynman amplitudes applies here as well. The analyticity of the amplitude is a conjecture, based on a proof that the second and third Born approximations satisfy a Mandelstam representation. The method of this proof is an adaptation of techniques recently developed by Eden and others in the relativistic case.