Exact L2 series solution of the Dirac-Coulomb problem for all energies

TL;DR

This paper presents an exact, series-based solution to the Dirac-Coulomb problem applicable to all energy levels, using hypergeometric functions and orthogonal polynomials to achieve a comprehensive analytical framework.

Contribution

It introduces a novel exact solution method for the Dirac-Coulomb problem valid for all energies, employing a tridiagonal matrix representation and special functions.

Findings

Exact solutions for all energies including discrete and continuous spectra.

Wavefunction expansion coefficients expressed via Meixner-Pollaczek polynomials.

Matrix representation of the Dirac-Coulomb operator is tridiagonal.

Abstract

We obtain exact solution of the Dirac equation with the Coulomb potential as an infinite series of square integrable functions. This solution is for all energies, the discrete as well as the continuous. The spinor basis elements are written in terms of the confluent hypergeometric functions and chosen such that the matrix representation of the Dirac-Coulomb operator is tridiagonal. The wave equation results in a three-term recursion relation for the expansion coefficients of the wavefunction which is solved in terms of the Meixner-Pollaczek polynomials.