Exact bound-state solution of a one-dimensional three-body system with aδinteraction

The one-dimensional three-body problem interacting with a pairwise $\delta$ function is investigated. For the general three-body problems, conditions on the masses and the interacting strengths for the existence of a simple type of exact solution are obtained. It is immediately clear that the "helium" atom does not possess such an exact solution. For the "helium" atom, an upper bound to the exact ground-state energy is obtained. This is close to, but slightly better than, the Hartree-Fock calculation which can be carried out exactly.