Non-Uniform Convergence of Two-Photon Decay Rates for Excited Atomic States

Two-photon decay rates in simple atoms such as hydrogen-like systems represent rather interesting fundamental problems in atomic physics. The sum of the energies of the two emitted photons has to fulfil an energy conservation condition, the decay takes place via intermediate virtual states, and the total decay rate is obtained after an integration over the energy of one of the emitted photons. Here, we investigate cases with a virtual state having an intermediate energy between the initial and the final states of the decay process, and we show that due to non-uniform convergence, only a careful treatment of the singularities infinitesimally displaced from the photon integration contour leads to consistent and convergent results