Calculation of Relativistic Single-Particle States

A computational method is proposed to calculate bound and resonant states by solving the Klein-Gordon and Dirac equations for real and complex energies, respectively. The method is an extension of a non-relativistic one, where the potential is represented in a Coulomb-Sturmian basis. This basis facilitates the exact analytic evaluation of the Coulomb Green's operator in terms of a continued fraction. In the extension to relativistic problems, we cast the Klein-Gordon and Dirac equations into an effective Schr\"odinger form. Then the solution method is basically an analytic continuation of non-relativistic quantities like the angular momentum, charge, energy and potential into the effective relativistic counterparts.