Expansion of the irregular solution in the theory of Stark effect in hydrogenic-like Rydberg atoms

TL;DR

This paper derives an expansion of the irregular solution in the Stark effect for hydrogenic-like Rydberg atoms, clarifying its relation to previous theories and providing analytical expressions for key matrices.

Contribution

It introduces a new expansion of the irregular solution at negative energies and links it to existing theoretical frameworks, enhancing the understanding of the Stark effect in Rydberg atoms.

Findings

The derived expansion is numerically equivalent to previous models at high angular momenta.

Analytical expressions for key matrices are provided.

The missing expansion in prior work is shown to follow from their main postulate.

Abstract

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.'s 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.