Separable Expansions of the Two-BodyTMatrix for Local Potentials and Their Use in the Faddeev Equation

Abstract

An expansion of the off-shell two-body $T$ matrix is introduced in the form of a sum of terms separable in the initial- and the final-momentum variables. The convergence of this expansion is tested against exact solutions of several local potential problems. The substitution of the expansion into the Faddeev equations yields a set of coupled integral equations in one variable. As an example, the binding energy of three identical bosons is calculated using an $S$-wave Yukawa potential for the two-body interaction. It is found that a stationary state exists for the potential strength $G\ensuremath{\ge}1.4$, whereas a two-body bound state requires $G\ensuremath{\ge}1.8$. An off-shell effective-range formula is introduced for problems in which the shape of the two-body potential is not well known. It is shown that the difference between the solutions of the Yukawa potential and the exponential potential with the same scattering length and effective range is comparable to the deviation of the off-shell effective-range formula from either solution.