Faddeev Equations for Atomic Problems. III. Convergence of the Separable-Expansion Method for Coulomb Interactions

The convergence of the separable-expansion method for solving the Faddeev equations is investigated for systems with pure Coulomb interactions between particles in each pair. Two alternative separable expansions for the off-shell two-body amplitudes are utilized. It is observed that the expansion method is capable of predicting qualitatively all the characteristic features such as the three-body bound-state and resonance poles, the scattering length, and the energy dependence of the phase shift in atomic systems such as the ($e$, H) system. Although some convergence behavior is observed, this method in the present form does not provide the desired accuracy when reasonable numbers of terms are included in the expansion for the attractive ($e-p$) scattering amplitudes. This behavior is interpreted as due to the failure of the two alternative expansions for the off-shell two-body amplitudes to converge monatonically.