L2 series solutions of the Dirac equation for power-law potentials at rest mass energy

TL;DR

This paper derives analytical solutions to the three-dimensional Dirac equation with power-law potentials at rest mass energy, expressing solutions as series of orthogonal polynomials for specific angular momentum and potential parameters.

Contribution

It introduces a method to obtain solutions as infinite series involving modified orthogonal polynomials, with a tridiagonal matrix representation of the Dirac operator.

Findings

Solutions expressed in terms of confluent hypergeometric functions.

Recursion relations solved using modified Meixner-Pollaczek and continuous dual Hahn polynomials.

Applicable to various power-law potentials at rest mass energy.

Abstract

We obtain solutions of the three dimensional Dirac equation for radial power-law potentials at rest mass energy as an infinite series of square integrable functions. These are written in terms of the confluent hypergeometric function and chosen such that the matrix representation of the Dirac operator is tridiagonal. The "wave equation" results in a three-term recursion relation for the expansion coefficients of the spinor wavefunction which is solved in terms of orthogonal polynomials. These are modified versions of the Meixner-Pollaczek polynomials and of the continuous dual Hahn polynomials. The choice depends on the values of the angular momentum and the power of the potential.