Effective Interactions for the Three-Body Problem

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 ($^{3}\mathrm{H}$ and $^{3}\mathrm{He}$), 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 $\ensuremath{\Lambda}$ are varied, including the case of a single included shell: $\ensuremath{\Lambda}\ensuremath{\hbar}\ensuremath{\omega}=0\ensuremath{\hbar}\ensuremath{\omega}$. For typical ranges of $b$, the induced effective three-body interaction, essential for giving the exact three-body binding, is found to contribute $\ensuremath{\sim}10%$ to the binding energy.