Determination of the Amplitude from the Differential Cross Section by Unitarity

Banach-space fixed-point theorems are used to prove two results: (a) If the differential scattering cross section is smooth and small enough, relative to the wavelength of the relative motion of the colliding particles, there always exists an amplitude function which satisfies elastic unitarity and whose squared modulus equals a given differential cross section. (b) Under somewhat stronger conditions this amplitude is uniquely determined (except for the sign of its real part) by the generalized optical theorem (unitarity) and it can be constructed by iterating the latter. The condition for (a) ensures a priori that the real part of the amplitude cannot vanish at any angle, and that for (b) implies that its real part cannot be smaller than twice its imaginary part. These results are then generalized to inelastic and production processes.