Two relativistic Sturmian basis sets are constructed by solving first-order Sturm-Liouville problems for the Dirac-Coulomb equation. The Sturmian expansions of the Dirac-Coulomb Green function and the reduced Dirac-Coulomb Green function are derived. Their utility is illustrated by calculating static electric dipole polarizabilities for hydrogen-like atoms with 1 Z 137.

Completeness of the Dirac oscillator eigenfunctions is proved in one and three spatial dimensions. Proofs are based on standard properties of the Hermite and the generalized Laguerre polynomials.

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The Sturmian expansion of the generalized first-order Dirac-Coulomb Green function (Szmytkowski 1997 J. Phys. B: At. Mol. Opt. Phys. 30 825, 2747) is used to derive an analytical formula for the static dipole polarizability of the relativistic hydrogen-like atom in the ground state. The formula contains a generalized hypergeometric series 3 F 2 with the unit argument. It is identical with one found recently, in a completely different way, by Yakhontov (2003 Phys. Rev. Lett. 91 093001) and is the most compact among all known analytical expressions for the polarizability. Partitioning of the polarizability into convection and spin-polarization components, resulting from the Gordon decomposition of an induced electronic charge density, is carried out. It appears that the spin-polarization component of the polarizability vanishes.