Asymptotic form of the electron capture cross section in the second Born approximation

The asymptotic variation of the amplitudes at high impact energies for the non-resonant H + + D(1s) -> H(1s) + D + and symmetrically resonant H + + H(1s) -> H(1s) + H + Proc.esses are calculated approximately in the second Born approximation using the final-state Green function G f Although agreement with the results of Drisko is obtained for scattering at small angles by approximating the Coulomb functions of G f by plane waves, this agreement may be fortuitous for two reasons (I) an exact calculation might yield ambiguous answers, (II) sums of contributions from an infinite number of discrete intermediate states, each varying with the same power of impact energy, may not be finite. In the case of symmetrical resonance not only do contributions also originate from the angular region near θ = π, but all but one of these calculated partial amplitudes is dominated by the corresponding first Born amplitude, however, the same calculational difficulties, (I) and (II), also occur for this case Contributions from the 1s intermediate state diverge for each part of the post interaction V f , but these divergences cancel when the complete interaction V f is used