Transition Matrix for Nucleon-Nucleon Scattering

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Abstract
As part of a study of the influence of off‐the‐energy‐shell effects on the optical potential for nucleon‐nucleus scattering, a method is presented for the calculation, via the reactance matrix, of the nucleon‐nucleon transition matrix in terms of an internucleon potential and the scattering amplitude. The singular integral equations for the partial‐wave amplitudes of the reactance matrix are reduced to a Fredholm form which contains the scattering amplitude parametrically. The iteration solution of these Fredholm equations is shown to be generally unreliable; however, the zeroth‐order iteration approximates the exact solution quite well near the energy shell. The replacement of the kernels of these integral equations by separable functions is discussed; the validity of such an approximation is illustrated by a simple example. The requirement that the solutions of the (exact) Fredholm equations be consistent with the original singular integral equations yields a solution for the scattering amplitude in terms of the resolvent kernels of the Fredholm equations. The entire formalism is so constructed as to include the possibility of a hard core being present in the nucleon‐nucleon interaction.

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