Transformation Between Regular-Irregular Schrödinger Equation Solution Pairs by Means of the Corresponding Milne Function Properties

In scattering and tunnelling-to-continuum problems, the regular and irregular solution pairs to the one-particle radial Schrödinger equation follow various amplitude and phase difference conventions at infinity. The Milne transformation of the Schrödinger equation (a.k.a. the Phase-Amplitude Method) leads to a solution pair whose form inherently complies with scattering theory’s requirements of common amplitude and phase difference of π/2 at the r→∞ limit. On occasion, applications favoured the adoption of the reverse phase convention. In this work we present a formal framework that permits switching of the regular/irregular solution pairs between representations. To showcase the formalism, two exemplar potential curves are investigated: a repulsive Coulomb barrier and the Coulomb-Stark potential, the latter corresponding to the hydrogen atom in the presence of a static homogeneous electric field. Analysis of the results, assessment of the methodology and a discussion of possible extensions are also included. Ultimately, this work shows that it is the regular solution that acts as the thread that binds together the different representations.