Solution of the singular Dirac equation using mapped trigonometric functions

This work was devoted to developing an effective spectral method for solving the Dirac equation with the singular Coulomb potential, where the fast Fourier transformation could be directly applied. In the method, the trigonometric sine functions were chosen as the basis, and a well-designed scaling function was proposed to account for the singularity of the Coulomb potential. In the spirit of the discrete variable representation, the Chebyshev-Gauss quadrature was adopted. It is found that the method could achieve spectral convergence easily to a precision around 10^{-32} with quadruple variables. Although the numerical solution of the Dirac equation is more demanding than the Schrödinger equation due to the relativistic effects, the numerical convergence of the proposed method is comparable with the best spectral method at the moment for solving the Dirac equation, the Lagrange mesh method.