Singularities of the partial-wave amplitudes in the three-body problem

The singularities of the partial-wave amplitudes for the diagrams of the multiple scattering series in a system of three nonrelativistic particles of arbitrary mass are investigated. For an arbitrary diagram, simple analytic expressions are obtained for the position of the nonpotential singularities of the partial-wave amplitudes in the energy plane and necessary and sufficient conditions for the existence of these singularities on the physical sheet are found. Simultaneously, using the analogy between the singularities of the diagrams of the multiple scattering series and the classical problem of the collision of three material points, we determine the maximal number of possible two-body collisions in a classical system of three point particles of arbitrary mass with contact interaction.