Three-dimensional formulation of the relativistic two-body problem in terms of rapidities

A scheme is developed which is based on the three-dimensional relativistic equation of the quasipotential type [1–4]. As a main variable the authors use the rapidity, i.e. the quantity which is canonically conjugated to the relativistic relative distance [5]. A free Green function has a simple pole in the complex plane of the rapidity, which guarantees the elastic unitarity in the case of a real potential. In the case of local potential, the corresponding equation for the partial wave function in the configurational $r$-representation is the second order differential equation. The question about formulating boundary conditions is investigated, which represents a nontrivial problem in the relativistic configurational space. Exact solutions of theequation in several simple cases are found.