Second-order eikonal approximation for scattering

An eikonal approximation for potential scattering at high energies is derived to second order in 1/k for the Schr\"odinger equation. The procedure is based on the same assumptions and restrictions introduced by Glauber, except that the derivation is carried one order further. Whereas Glauber's solution provides for phase modulation of the incoming plane wave, the second-order eikonal approximation, in addition to amending the phase term, turns out also to include an amplitude modulation factor. This approximation, which is the result of maintaining the original calculation to higher order, differs in general from the perturbation correction by Wallace of Glauber's first-order eikonal approximation, although both reduce to the same correct solution for the special case of Coulomb scattering.