Quasirelativistic Coulomb Sturmian basis sets for atomic physics problems: illustration of their applicability to two-photon decay rates in hydrogen-like atoms

The continuum and discrete quasirelativistic one-electron Sturmian basis sets associated with the second-order Dirac-Coulomb equation are constructed by solving appropriate Sturm-Liouville problems. The potential-weighted orthogonality and closure relations obeyed by the members of such sets are discussed. The special case where both quasirelativistic Coulomb Sturmian basis sets are used to obtain solutions to the Gell-Mann-Feynman equation is presented. It is proved that for the continuum quasirelativistic Sturm-Liouville problem, the spectrum of potential strengths is continuous and covers the whole real axis. We have also introduced the complete outgoing continuum quasirelativistic Coulomb wavefunctions and presented the discretization of its radial components. The compact representation of the fully relativistic and quasirelativistic Coulomb Green's functions is outlined. By making use of the Sturmian expansion of the Coulomb Green's functions, closed-form expressions of two-photon spontaneous decay rates are derived for arbitrary multipole channels. For the sake of assessing the domain of applicability of quasirelativistic basis sets, an illustrative application is then carried out for two-photon decay rates for hydrogen-like ions. A comparison of present results with reference values previously obtained within a fully relativistic scheme (Bona et al 2014) is undertaken.