Two-body relativistic scattering of directly interacting particles

Two-body relativistic scattering via direct interaction is discussed. For a certain class of interactions the problem is equivalent to a family of nonrelativistic scattering problems. The relativistic mass shell is shown to correspond with the kinetic-energy shell of the reduced problem. Under certain sufficient assumptions it is shown that known proofs apply, which ensure that the wave operators exist off-shell. Under more precise conditions, the scattering operator $S$ can be defined also on the mass shell, which is stable by its action. Moreover, $S$ preserves the positivity of energy and is unitary with respect to the inner product of on-shell states. In a more particular case, we obtain a natural normalization in the interacting mass shell.