Low-Energy Electron Scattering in the Random-Phase Approximation

A general formalism for the computation of low-energy inelastic and elastic electron scattering cross sections within the context of the particle-hole Bethe-Salpeter equation is presented and shown to reduce to the random-phase approximation (RPA) in lowest order. The theory is then applied to triplet elastic electron-${\mathrm{He}}^{+}$ scattering. A short discussion of the differences between the RPA for scattering processes and for low-lying bound states is given, and the numerical methods used to solve the equations are considered in some detail in an appendix.