Determination of the scattering amplitude (Q2266407)

Within the range of elastic scattering of two particles the unitarity equation can be formulated as an integral equation for the phase of the analytic scattering amplitude if the cross-section is given. Since the publications of \textit{R. G. Newton} [Determination of the amplitude from the differential cross-section by unitarity, J. Math. Phys. 9, 2050-2055 (1968)] and \textit{A. Martin} [Construction of the scattering amplitude from the differential cross-sections, Nuovo Cimento 59 A, 131-152 (1969) and Scattering theory: Unitarity, analyticity and crossing, Lect. Notes Phys. 3 (1969)] this nonlinear equation has been investigated as a fixed- point problem in a real Banach space with the Banach contraction mapping principle or the Schauder fixed-point theorem. In the present publication this mapping is reexamined in a complex Banach space and the real integral equation is extended to a fixed-point problem in a complex Banach space. The fixed point theorem of Earle and Hamilton for holomorphic mappings leads then to the determination of a unique scattering amplitude if the parameter sin \(\mu\) of Newton and Martin (which characterizes the cross-section) is bounded by the improved limit sin \(\mu\) \(<0,86\).