Gauge invariance and approximate multiphoton calculations in hydrogen

It has been observed that fairly accurate estimates for two-photon bound-bound amplitudes in hydrogen can be obtained from a simple partial summation restricted to the discrete spectrum contribution, provided one uses the E·r form of the interaction Hamiltonian. We show that this conclusion does not hold when considering higher-order multiphoton processes. In fact, the corresponding multiple sums can be shown to be divergent and, consequently, no reliable estimates can be obtained from such an approximation. Using the (1/c)A·p form of the interaction does not lead to reliable estimates either. Those conclusions are based on a comparison with exact results for the second- and fourth-order contributions to the ac Stark shift of the ground state, computed with the help of a Sturmian basis