Smooth, invariant orthonormal basis for singular potential Schroedinger operators

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.