The analyticity of the scattering amplitude in angular momentum for $N$-coupled relativistic two-body channels is investigated on the basis of Mandelstam representation and unitarity. The problem of the proof of the analytic properties of the amplitude is reduced to the boundedness of a particular kernel involving the left-hand discontinuity of the amplitude. The behavior of the Regge trajectories at inelastic thresholds is determined. The results are extended to relativistic models with infinite-dimensional unitarity relation but without crossing symmetry such as the Bethe-Salpeter amplitude. The implications of the results to the exact $S$-matrix theory are also discussed.

• Received 25 June 1962

DOI:https://doi.org/10.1103/PhysRev.128.1959