Multiphoton ionization by intense fields: Corrections to lowest order perturbation theory, with an application to photoionization of H

We consider the perturbation expansion (in powers of the coupling strength of the electron interaction with the radiation field) of the standard time-independent expression for the multiphoton ionization amplitude. We show that corrections beyond lowest nonvanishing order contain divergent diagrams. These divergences cancel in the velocity gauge but not in the length gauge. We discuss the mathematical and physical nature of these divergences, and we present an application to the photoionization of hydrogen.