  Schr\"odinger operators often display singularities at the origin, the
Coulomb problem in atomic physics or the various matter coupling terms in the
Friedmann-Robertson-Walker problem being prominent examples. For various
applications it would be desirable to have at one's disposal an explicit basis
spanning a dense and invariant domain for such types of Schr\"odinger
operators, for instance stationary perturbation theory or the Raleigh-Ritz
method.
  Here we make the observation, that not only a such basis can indeed be
provided but that in addition relevant matrix elements and inner products can
be computed analytically in closed form, thus providing the required data e.g.
for an analytical Gram-Schmid orthonormalisation.
