Asymptotic behavior of charge-exchange amplitude at relativistic velocities and binding energies

The leading asymptotic term (for large impact parameters) of the charge-exchange amplitude for $1s_{1/2}\to 1s_{1/2}$ transitions is calculated in a modified relativistic eikonal approximation. Relativistic effects are taken into account in the kinematics of electron capture and in the description of the bound states. A study is made of the effects of the long-range interaction due to the Coulomb interaction between the particles in the entrance and exit reaction channels. The analytic results obtained in the nonrelativistic limit ($c\to \infty$) for the charge-exchange amplitude in the asymptotic region of large impact parameters are valid for all velocities, going over in the adiabatic limit into the well-known expressions of Firsov and in the limit of high velocities into the expressions of the Born approximation.