On the Application of Faddeev Equations and the CoulombtMatrix to Asymptotic Electron Capture in Hydrogen

Faddeev equations together with the Coulomb $t$ matrix have been used to determine the asymptotic amplitude for electron capture from neutral hydrogen by fast protons. The results show that in the high-energy limit the capture cross section should go down as $v^{-11\mathrm{}}$, where $v$ is the velocity of the incident proton. The capture amplitude is identical to Drisko's second-Born-approximation calculation except for a complex energy-dependent phase factor which ultimately approaches unity with sufficiently high incident energy. The major contribution to the three-body capture amplitude can be shown to come from the on-energy-shell two-body $t$ matrix, in agreement with general theorems concerning scattering from complex systems. At high incident energies, the on-energy-shell contribution to the capture amplitude (not the cross section) will decrease as $v^{-5\mathrm{}}$, while the off-energy-shell continuum contribution will decrease as $v^{-6\mathrm{}}$. The contributions from the sum of the infinite number of two-body bound-state poles can be shown to converge, and the sum can be explicitly performed at high enough incident energies in all except the forward direction. The bound-state contributions to the capture amplitude go down as $v^{-11\mathrm{}}$, which is much less than the continuum contributions.