A test of approximation methods in potential scattering

Abstract

Various approximation procedures fashionable in field theory for computing scattering amplitudes are tested on a soluble problem in potential theory — thes-wave scattering by an exponential potential. The methods include: 1) the Fredholm, or determinantal, expansion; 2) the Chew-Mandelstam procedure of constructing the scattering amplitudeT0 from analyticity properties and unitarity; 3) expansion ofT0 in powers of the potential strengthλ (Born approximation); 4) expansion of tgδ0 in powers ofλ, and 5) expansion of ctgδ0 in powers ofλ. Each is carried out in first and second order of approximation and compared with the exact result. The results are displayed in effective-range plots ofk ctgδ0vs energy. In addition, the energies of bound states as predicted by the approximations are compared with the exact result. Approximations 1), 2), and 5) in second order are comparable in accuracy, agree reasonably well with the exact result, and are appreciably better than 3) and 4). The binding energy of the first bound state is predicted well by method 2) in second order, and at best qualitatively by the other methods. All methods except 1) predict existence of bound states for repulsive potentials. In second order method 1) predicts no bound state for any value ofλ.