Uniqueness of the Partial-Wave Amplitudes

The following result is proved: Given all but a finite number of partial-wave amplitudes in a two-body scattering process, the remaining amplitudes are uniquely determined if either (a) the scattering amplitude has crossing symmetry or (b) there is an energy region in one of the crossed channels where the scattering is purely elastic. In case (a) the proof does not require a knowledge of the precise analytic structure of the scattering amplitude, while in case (b) the amplitude is assumed to satisfy the Mandelstam representation.