Coulomb divergence in S-matrix expansion of above-threshold ionization

Photoelectron spectrum of above-threshold ionization (ATI) is calculated using the first two orders of Coulomb-Volkov S-matrix series (Faisal, Phys. Rev. A 94, 031401 (2016)). Calculation shows that Coulomb divergence, due to singularity of forward scattering, still exists in the first two order terms for an unscreened Coulomb potential. This divergence can be removed by adding a decay factor into the phase of the S-matrix element which comes from depletion of the atomic ground state. We show that by using an example of hydrogen atom, the Coulomb-Volkov S-matrix series can successfully describe the energy spectrum of ATI and high-order ATI with the introduction of a deplete rate of the ground state.