First and second bound state for a three-body system assuming a local two-body potential with repulsive term

Abstract

In this paper we study the Faddeev equations for a system of three identical and spinless particles, assuming a local two-body potential tha t contains a short-range repulsive term. We shall deal with the problem of the determinat ion of the binding energies of the first and second bound state of the system. Though many calculations on the Faddeev equations with local potentials (1) are by now available, there are still unsolved problems in the numerical methods of solution. The method we have used provides an efficient and mathematical ly well-founded technique, suitable also for the determinat ion of the second bound state. This point was not yet investigated for the difficulties arising near the energy threshold. The approach is the same as in ref. (2), where the authors were interested in the determinat ion of the first three-body bound state, taking as two-body interaction an attractive Yukawa potential. As regards the equations we deal with and the method used for their solution, we make reference to the paper in ref. (2) (in the following denoted by I) and we spend only few words in order to make self-contained a first reading of this paper. We make use of variational technique, tha t exploits some precise connections with the Pad~-approximants (PA) method (~), in order to obtain the binding energies of a three-body system. To go further into details, the Faddeev equations (4) are rewritten formally in the same way as the Lippman-Schwinger equat ion for the off-shell two-body scattering ampli tude (I, eqs. (2), (3)). Introducing a fictitious parameter 2, which has to be set equal to one at the end of the calculations, it is possible to write a Neiimann expansion oa ~, for the diagonal matr ix elements of the off-shell scattering amplitude (I, eq. (8)).