Perturbative and nonperturbative calculations of electron-hydrogen ionization

We compare calculations of the fully differential cross section for ionization of atomic hydrogen by electron impact using two different theories—the perturbative CDW-EIS (continuum distorted wave with eikonal initial state) approximation and the nonperturbative ECS (exterior complex scaling) method. For this comparison, we chose an impact energy of 54.4 eV, since this is near the lowest energy that our perturbative approach would be applicable and near the highest energy that can be tackled by the ECS method with our present computational resources. For the case of equal-energy outgoing electrons investigated here, the two theories predict nearly identical results except that CDW-EIS underestimates the ECS values nearly uniformly by about 30%. Interestingly, when initial-state projectile-target interactions are neglected by replacing the eikonal initial state with the unperturbed initial state (the approximation of Brauner, Briggs, and Klar [J. Phys. B $22,$ 2265 (1989)]), the cross section oscillates by $\pm 50\%$ about the ECS values.