Representation of solutions of the one-dimensional Dirac equation in terms of Neumann series of Bessel functions

TL;DR

This paper presents a novel representation of solutions to the one-dimensional Dirac equation using Neumann series of Bessel functions, enabling efficient numerical computation of spectral data with high accuracy.

Contribution

It introduces a new uniform convergence representation of Dirac solutions via Bessel functions and provides explicit recursive formulas for coefficients, enhancing numerical methods.

Findings

The solutions are uniformly convergent with respect to the spectral parameter.

Explicit formulas for coefficients are derived using recursive integrals.

The method efficiently computes large eigendata sets with stable accuracy.

Abstract

A representation of solutions of the one-dimensional Dirac equation is obtained. The solutions are represented as Neumann series of Bessel functions. The representations are shown to be uniformly convergent with respect to the spectral parameter. Explicit formulas for the coefficients are obtained via a system of recursive integrals. The result is based on the Fourier-Legendre series expansion of the transmutation kernel. An efficient numerical method for solving initial-value and spectral problems based on this approach is presented with a numerical example. The method can compute large sets of eigendata with non-deteriorating accuracy.