CoulombTMatrix and the Proton-Hydrogen Charge Exchange

The proton-hydrogen charge-exchange amplitude is evaluated using the first-order terms in the Faddeev expansion for the corresponding transition operator. The new Coulomb $T$ matrix derived by us recently is employed in computing the cross sections. For high energies of the incident proton, the Coulomb $T$ matrix approaches the Coulomb potential to order $v^{-1\mathrm{}}$ $\ln v$ in contrast to the situation for a short-range potential ($\sim v^{-2\mathrm{}}$). In the extreme high-energy limit the results of the first-order Faddeev-Watson approximation approach those of the Jackson-Schiff approximation. Explicit numerical calculation in the energy region 100 keV to 3 MeV shows that the cross sections lie in between those of the Brinkman-Kramers and Jackson-Schiff (JS) results and they approach the JS results from above. This is in contradistinction to our previous work based on an incorrect form of the $T$ matrix.