The Runge-Lenz operator enables an algebraic solution of Coulomb potential energy levels without a solution of a differential equation. What is the analog for the solution of the Dirac equation in a Coulomb potential and the associated energy levels. Is it possible to get the energy levels solely by algebraic methods and not solving the coupled differential equations. I have seen separation into angular and radial parts by algebraic means, but the radial equation always seems to be solved via the usual power series methods. Can this be avoided by using an analog of the Runge-Lenz operator for the Dirac equation?
L I Komarov and T S Romanova 1985 J. Phys. B: At. Mol. Phys. 18 859 The algebraic method of solution of the Dirac equation for a particle in a Coulomb potential
Abstract:An equation is constructed in two-dimensional complex space, in the set of solutions of which solutions of the Dirac equation for a particle in a Coulomb potential are present. These solutions are found by a purely algebraic method.
I have not seen the article though.