Historical pseudo simplified solution of the Dirac-Coulomb equation

TL;DR

This paper critically examines a 1985 simplified solution to the Dirac-Coulomb equation, revealing that the introduced parameters and transformations are mathematically inconsistent and the solution cannot be reliably recovered.

Contribution

It provides a detailed critique showing the original solution's mathematical flaws and invalidates its claimed eigenvalues and transformations.

Findings

The original solution's parameters are mathematically inconsistent.

The transformation formulas used are self-contradictory.

The eigenvalues set violates solution uniqueness.

Abstract

One of the simplified solutions of the Dirac equations with the pure Coulomb potential given in a paper published in 1985 is pseudo. The original paper solved the Dirac equations by introducing a transformation of functions with two strange parameters a and b to transform the original system of the first-order differential equations into two uncoupled differential equations of second order. However, not only the given eigenvalues sets violate the uniqueness of solution but also the said second-order equations are not any necessarily mathematical deduction. In order to determine the introduced parameters, formally, the author actually introduced some self-contradictory mathematical formulas, such as sinh(theta)=2ab, cosh(theta)=a^2+b^2, tanh(theta)=-Z(alpha)/k, a^2-b^2=1, b=sinh((theta/2), a=cosh((theta/2). But one has not known the value of the parameters a and b all the while, whereas the parameters were insensibly deleted in the given second-order Dirac-Coulomb equation last. One cannot recover any result given in the paper by making corresponding correctly mathematical calculations.