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

Abstract

While the square root of Dirac's $\delta$ is not defined in any standard mathematical formalism, postulating its existence with some further assumptions defines a generalized function called $\gamma \left(x\right)$ 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 $\gamma \left(x\right)$ 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 $\gamma \left(x\right)$ generates several paradoxical situations which are collected here. The help of the scientific community is sought to resolve these paradoxa.

Keywords: Gamma function; Kinetic postulate; Square root of Dirac-delta.