Non-uniform convergence of two-photon decay rates for excited atomic states

TL;DR

This paper investigates the convergence properties of two-photon decay rates in hydrogen-like atoms, revealing that careful handling of singularities is essential for obtaining consistent results in cases with intermediate virtual states.

Contribution

It demonstrates the importance of non-uniform convergence and the need for careful treatment of singularities in calculating two-photon decay rates with intermediate virtual states.

Findings

Non-uniform convergence affects decay rate calculations.

Proper treatment of singularities yields consistent results.

Intermediate virtual states require careful mathematical handling.

Abstract

Two-photon decay rates in simple atoms such as hydrogenlike systems represent rather interesting fundamental problems in atomic physics. The sum of the energies of the two emitted photons has to fulfill 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 energy intermediate between the initial and the final state 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.