  In a recent contribution we showed that there exists a smooth, dense domain
for singular potential Schr\"odinger operators on the real line which is
invariant under taking derivatives of arbitrary order and under multiplication
by positive and negative integer powers of the coordinate. Moreover, inner
products between basis elements of that domain were shown to be easily
computable analytically.
  A task left open was to construct an orthonormal basis from elements of that
domain by using Gram-Schmidt orthonormalisation. We perform that step in the
present manuscript. We also consider the application of these methods to the
positive real line for which one can no longer perform the integrals
analytically but for which one can give tight analytical estimates.

Quantum Physics;General Relativity and Quantum Cosmology;High Energy Physics - Theory