Effective interactions for the three-body problem -: art. no. 014316

The three-body energy-dependent effective interaction given by the Bloch-Horowitz equation is evaluated for various shell-model oscillator spaces. The results are applied to the test cases of the three-body problem (H-3 and He-3), where it is shown that the interaction reproduces the exact binding energy, regardless of the parametrization (number of oscillator quanta or value of the oscillator parameter b) of the low-energy included space. We demonstrate a nonperturbative technique for summing the excluded-space three-body ladder diagrams, but also show that accurate results can be obtained perturbatively by iterating the two-body ladders. We examine the evolution of the effective two-body and induced three-body terms as b and the size of the included space Lambda are varied, including the case of a single included shell: Lambda(h) over bar omega=0 (h) over bar omega. For typical ranges of b, the induced effective three-body interaction, essential for giving the exact three-body binding, is found to contribute similar to10% to the binding energy.