Extension of the Schwinger variational principle beyond the static-exchange approximation

We propose a new vairational principle for scattering theory which extends the Schwinger variational principle beyond the static-exchange approximation and to inelastic scattering. Application of this formulation to the scattering of electrons by hydrogen atoms at energies below $k^2=0.64\mathrm{}$ demonstrates the rapid convergence of the phase shift with respect to the number of basis functions for both the open- and closed-channel orbitals. Furthermore, we show that the convergence of the phase shift with respect to the number of expansion functions (exact states or pseudostates) is also fast. In our theory, the resulting phase shifts can be more accurate than those of the close-coupling method even if the same expansion basis is used. The phase shifts in our $1s-\overline{2s}-\overline{2p}$ calculation are comparable to those of $1s-2s-2p-\overline{3p}-\overline{3d}$ calculation of Matese and Oberoi [Phys. Rev. A 4, 569 (1971)], which are very close to the exact values. Several aspects of the convergence characteristics are also discussed.