  The usefulness of the Relativistic Schr\"odinger Theory (RST) is studied in
the field of atomic physics. As a concrete demonstration, the positronium
groundstate is considered in great detail; especially the groundstate energy
$E_0$ is worked out in the non-relativistic approximation and under neglection
of the magnetic interactions between the positron and the electron. The
corresponding RST prediction $(E_0\simeq -6,48 [eV])$ misses the analogous
conventional Schr\"odinger result $(E_0\simeq -6,80 [eV])$ but is closer to the
latter than the corresponding Hartree approximation $(-2,65 [eV])$. The missing
binding energy of $6,80-6,48=0,32 [eV]$ can be attributed to the approximative
use of an SO(3) symmetric interaction potential which in RST, however, is
actually only SO(2) invariant against rotations around the z-axis. It is
expected that, with the correct use of an anisotropic interaction potential due
to the SO(2) symmetry, the RST predictions will come even closer to the
conventional Schr\"odinger result, where however the mathematical structure of
RST relies on exotic (i.e. double-valued) wave functions and on the
corresponding unconventional interaction potentials (e.g. Struve-Neumann
potential).
