Faddeev approach to atomic three-body problems

The Faddeev equations for the (e⁻,H) system have been solved numerically using separable expansions for both screened and unscreened versions of the Coulomb t matrix. By considering the equations in uncoupled form, the convergence behavior of the binding energy and the singlet scattering length have been examined as the number of terms in the representation of the electron-electron potential has been increased. One to three states have been included in the e-p interaction and up to twenty states in the e-e interaction. Although the unscreened values are reasonable in low orders of truncation, the convergence is poor as more terms are included in the separable representations. To improve the rate of convergence, a screened form of the Coulomb t matrix based on the Sturmian expansion of the Hulth\'en potential has been introduced. In the lowest trucation of the e-p system, strong shielding of the e-e interaction is necessary and convergence of both the binding energy and the scattering length can be achieved. In higher truncations of the e-p system, where the representation of the e-p interaction is more realistic, shielding is not effective in improving the rate of convergence within the practical limits of the calculations. The implications of the present calculations for the Faddeev approach to atomic three-body problems are discussed.