Numerical split-shift potential method for relativistic quantum systems with radial symmetry

We show how the spectrum of a radially symmetric Dirac-Hamiltonian can be computed rather accurately on a spatial grid based using a split-shift potential method. This method is sufficiently accurate such that the fine structure splittings of hydrogen-like relativistic ions with nuclear charge Z can be reproduced for a relatively small number of spatial radial grid points. We use this analytically known spectrum to examine the error scaling of this method. The method is then applied to examine the impact of a spatial confinement on the fine structure splittings and the bound states for hydrogen.