Fredholm Method. II. A Numerical Procedure for Inelastic Scattering

A convenient and accurate numerical method is given whereby inelastic scattering information can be obtained by construction of the Fredholm determinant det[1-G(E+iε)V] for the coupled Lippmann-Schwinger equations. The method is noniterative and is easily applied when the potential matrix is nonlocal or energy dependent. It is shown that the determinant det[1-G(E+iε)V] may be factored as det[1-PG(E)V]det(1-iR) when PG(E) is the principal-value Green's function and R is the usual R matrix of principal-value Lippmann-Schwinger theory; the R matrix may be obtained from det[1-G(E+iε)V] by a single partial triangularization. As a simple example of the extraction of the R matrix from the Fredholm determinant, the problem of electron scattering from hydrogen atoms is considered in the $1s$, 1s-2s, 1s-2s-3s, and 1s-2s-3s-4s close-coupling approximations. The use of optical potentials in the Fredholm theory is discussed: The two-channel problem originally suggested by Huck is solved numerically by construction of an optical potential.