Expansion of the Irregular Solution in the Theory of Stark Effect in Hydrogenic-Like Rydberg Atoms

We derive the expansion of the irregular physical solution over the spherical solutions at negative energies, which is necessary for obtaining the S matrix of the process. The relation of this expansion to the theory developed by Giannakees et al., Phys. Rev. A 94, 013419 (2016), is analyzed. In particular, we show that the expansion of the irregular solution missing in Giannakees et al. theory can be derived from one of their main postulates. The expansion thus obtained turns out to be numerically equivalent to our expansion up to high angular momenta. Analytical expressions for the key matrix of both expansions are derived.