Equivalence of the generalized and complex Kohn variational methods

TL;DR

This paper proves the equivalence of the complex and generalized Kohn variational methods for low energy positron scattering, providing a general framework applicable to various potential scattering problems and explaining singularity behaviors.

Contribution

It establishes a formal equivalence between the complex and real generalized Kohn methods and introduces a framework to analyze singularities without explicit solutions.

Findings

Proves the phase shift from complex Kohn equals that from real generalized Kohn.

Develops a framework to describe and analyze Schwartz singularities.

Shows explicit solutions of Kohn equations are unnecessary for optimal phase shift calculations.

Abstract

For Kohn variational calculations on low energy positron hydrogen molecule elastic scattering, we prove that the phase shift approximation obtained using the complex Kohn method is precisely equal to a value which can be obtained immediately via the real-generalized Kohn method. Our treatment is sufficiently general to be applied directly to arbitrary potential scattering or single open channel scattering problems, with exchange if required. In the course of our analysis, we develop a framework formally to describe the anomalous behaviour of our generalized Kohn calculations in the regions of the well known Schwartz singularities. This framework also explains the mathematical origin of the anomaly-free singularities we reported in a previous article. Moreover, we demonstrate a novelty, that explicit solutions of the Kohn equations are not required in order to calculate optimal phase shift approximations. We relate our rigorous framework to earlier descriptions of the Kohn-type methods.