S-Matrix Method for the Numerical Determination of Bound States

A rapid numerical technique for the determination of bound states of a partial-wave-projected Schrödinger equation is presented. First, one needs to integrate the equation only outwards as in the scattering case, and second, the number of trials on ${\kappa }^2 \mathrm{}(=-E)\mathrm{}$ necessary to determine the eigenenergy and the corresponding eigenfunction is considerably less than in the usual method. As a nontrivial example of the technique, bound states are calculated in the exchange approximation for the $e^{-}-{\mathrm{He}}^{+}$ system and $l=1\mathrm{}$ partial wave.