  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.
