The Dirac-Coulomb Problem: a mathematical revisit

TL;DR

This paper presents a mathematical analysis of the Dirac-Coulomb problem, deriving a matrix representation and using orthogonal polynomial techniques to compute bound states, scattering amplitudes, and phase shifts.

Contribution

It introduces a symmetric tridiagonal matrix representation of the Dirac-Coulomb operator and applies orthogonal polynomial methods to analyze its spectral properties.

Findings

Derived bound state energy spectrum

Calculated relativistic scattering amplitudes

Determined phase shifts from polynomial asymptotics

Abstract

We obtain a symmetric tridiagonal matrix representation of the Dirac-Coulomb operator in a suitable complete square integrable basis. Orthogonal polynomials techniques along with Darboux method are used to obtain the bound states energy spectrum, the relativistic scattering amplitudes and phase shifts from the asymptotic behavior of the polynomial solutions associated with the resulting three-term recursion relation.