Convergence of anL2approach in the coupled-channel optical-potential method fore-H scattering

An ${\mathit{L}}^2$ approach to the coupled-channel optical method is studied. The investigation is done for electron-hydrogen elastic scattering at projectile energies of 30, 50, 100, and 200 eV. Weak-coupling, free-particle Green’s function, and no exchange in Q space are approximations used to calculate the polarization potential. This model problem is solved exactly using actual hydrogen discrete and continuum functions. The convergence of an ${\mathit{L}}^2$ approach with the Laguerre basis to the exact result is investigated. It is found that a basis of ten Laguerre functions is sufficient for convergence of approximately 5% in the polarization potential matrix elements, and 2% in the differential cross sections for nonlarge angles. The convergence is faster for smaller energies. In general, the convergence to the exact result is slow.