The function in quantum theory II. Mathematical challenges and paradoxa.

While the square root of Dirac's is not defined in any standard mathematical formalism, postulating its existence with some further assumptions defines a generalized function called which permits a quasi-classical treatment of simple systems like the H atom or the 1D harmonic oscillator for which accurate quantum mechanical energies were previously reported. The so-defined is neither a traditional function nor a distribution, and it remains to be seen that any consistent mathematical approaches can be set up to deal with it rigorously. A straightforward use of generates several paradoxical situations which are collected here. The help of the scientific community is sought to resolve these paradoxa.