A quantum-mechanical problem of a pair of identical particles interacting with a third particle via zero-range interactions is discussed.
A complete description is given to a system of two identical fermions (bosons) and a third particle. Correct formulation of the problem is presented, and bound-state energies are calculated for various total angular momenta. For this purpose, the generalized Coulomb problem is analyzed and its correspondence to the three-body problem is used to introduce the boundary condition in the vicinity of the triple-collision point. The related mathematical aspects and remaining obscure points are discussed.
Exact diffractionless solutions are obtained for the system of two-component one-dimensional particles with zero-range interactions. A new three-body bound state of odd parity is found, and the region of the parameters for which it exists is identified. One of the boundaries of this region is exactly determined by a diffractionless solution.
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