Static Green's Function for Elastic Electron-Hydrogen Scattering and Resonances

One version of the variational bound formulation of scattering theory requires the evaluation of integrals involving the Green's function GP(E) for the static, or one-body, approximation in which there is a hydrogen atom in the ground state throughout the scattering process. We obtain explicit expressions for GP(E) for the problem of the elastic scattering of electrons by atomic hydrogen, for both singlet and triplet states and for arbitrary total orbital angular momentum, in terms of solutions of integrodifferential equations. The triplet $L=0$ case requires special treatment because the hydrogenic ground-state function is a solution of the homogeneous (static) equation; this makes the integrodifferential operator, as it stands, singular, and the operator must be modified. In all cases, the integrodifferential equations are transformed into integral equations which are solved numerically. The numerical evaluation of GP(E) also enables one to determine the resonance energies for scattering as the eigenvalues of a (nonlinear) operator. The method developed is applicable to a number of other problems.