Calculable methods for many-body scattering

The review consists of two major parts. In the first part, several calculable $R$-matrix and related theories are described and discussed. These include the Kapur-Peierls, Wigner-Eisenbud, calculable standard $R$-matrix, extended $R$-matrix, finite-element, natural boundary condition, and variational methods. The various approaches are critically compared using four selected applications: (i) exactly soluble model using two coupled square-well potentials, (ii) elastic scattering of neutrons from ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$, (iii) elastic scattering of electrons from He atoms, and (iv) $\alpha -\alpha$ elastic scattering. In the second part, the Baer, Kouri, Levin, and Tobocman many-body scattering theory is reviewed. The principal results of the theory are derived, and a survey of calculations applying the theory is presented. The derivation is carried out in the context of the $R$-matrix method wherein the many-body scattering is treated ab initio as a steady-state process. This has the advantage that the channel states form a complete orthogonal set. These same channel states are used to provide explicit representations of the partition Green's-function operators.