Extrapolation to the Limit of Singular-Core Interactions in the Three-Body Problem

Several aspects of the three-body problem with singular-core interactions given by the boundary-condition model (BCM), are studied. The kernel of the Faddeev equations for these interactions is shown to have an infinite Schmidt norm even if the two-body interactions are confined to a single partial wave. This does not necessarily imply that the Faddeev kernel is noncompact. A proof of the compactness (or noncompactness) of the kernel has not been found. We consider a family of two-body interactions, with square repulsions of strength $V$ for particle separation $r<\mathrm{}{r}_0$ which become the BCM in the limit $V_0\to \infty$. For finite $V_0$ the Faddeev kernel has a finite Schmidt norm (and is hence compact) and standard numerical matrix texhniques may be used for solving the three-body equations. Simplified calculations of the triton binding energy, using two-body $s$-wave interactions of this type, are carried out for a number of choices of $V_0$. The two-body potentials for $r>\mathrm{}{r}_0$ are not varied. It is found that the three-particle binding energy has a simple dependence on $V_0$. The value of the binding energy extrapolated to the limit $V_0=\infty$ is found to be in excellent agreement with the result of a previous calculation based on BCM two-particle interactions and numerical methods predicated on the assumed validity of standard matrix-inversion techniques. Some implications of these results for more realistic calculations on three-body systems with two-body singular-core interactions are discussed.