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 $\det [1-\mathit{G}(E+i\epsilon \left)\mathit{V}\right]$ 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 $\det [1-\mathit{G}(E+i\epsilon \left)\mathit{V}\right]$ may be factored as $\det [1-\mathcal{P}\mathit{G}\mathrm{}(E\left)\mathrm{}\mathit{V}\right]\det (1-i\mathit{R})$ when $\mathcal{P}\mathit{G}\mathrm{}\left(E\right)\mathrm{}$ 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 $\det [1-\mathit{G}(E+i\epsilon \left)\mathit{V}\right]$ 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.