Determination of the scattering amplitude from the differential cross section for scattering by an arbitrary noncentral potential

The unitarity relation of quantum-scattering theory is studied for the general case of scattering by an arbitrary noncentral force and it is shown how it can be utilized to determine the phase of the scattering amplitude from the measured differential cross section. It is found that the unitarity relation leads to a pair of nonlinear integral equations for the phase of the scattering amplitude if the cross section is known. It is shown that from these equations one can obtain, without introducing approximations, a system of linear algebraic equations for a certain set of scalars from the values of which one can readily calculate the phase. In general, the above system of linear equations may admit also redundant nonphysical solutions for the phase. However, when the magnitude of the cross-section data satisfies certain conditions the solution of the equations is unique and must equal the correct physical value of the phase. Unlike previous methods for phase determination, the present approach is based on a form of the unitarity equation which is not simplified by any of the assumptions of parity, rotational, or time-reversal invariance for the underlying scattering interaction. Therefore, it can be applied to the determination of the phase also in cases such as particle scattering by, or in the presence of, external fields which do not possess the above-mentioned symmetries. A generalization of this method to many-channel scattering is also provided. Similar results hold also for the determination of phases in the theory of electromagnetic-wave scattering by an obstacle of arbitrary shape.