Stationarity of resonant pole trajectories in complex scaling

A reciprocity theorem relating the real parameters η and α that define the complex scaling transformation r →η re i α in the theory of complex scaling for resonant states is demonstrated. The virial theorem is used in connection with the stationarity of the pole trajectory. The Stark broadening in the hydrogen atom using a basis set generated by Rayleigh–Schrödinger perturbation theory is treated as an example.