Effective interactions for the three-body problem

The three-body energy-dependent effective interaction given by theBloch-Horowitz (BH) equation is evaluated for various shell-model oscillatorspaces. The results are applied to the test case of the three-body problem(triton and He3), where it is shown that the interaction reproduces the exactbinding energy, regardless of the parameterization (number of oscillator quantaor value of the oscillator parameter b) of the low-energy included space. Wedemonstrate a non-perturbative technique for summing the excluded-spacethree-body ladder diagrams, but also show that accurate results can be obtainedperturbatively by iterating the two-body ladders. We examine the evolution ofthe effective two-body and induced three-body terms as b and the size of theincluded space Lambda are varied, including the case of a single includedshell, Lambda hw=0 hw. For typical ranges of b, the induced effectivethree-body interaction, essential for giving the exact three-body binding, isfound to contribute ~10% to the binding energy.Comment: 19 pages, 9 figures, submitted to PR