2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/208116The inner-outer part factorisation of analytic representations in the unit disk is used for an effective characterisation of the number-phase statistical properties of a quantum harmonic oscillator. It is shown that the factorisation is intimately connected to the number-phase Weyl semigroup and its properties. In the Barut-Girardello analytic representation the factorisation is implemented as a convolution. Several examples are given which demonstrate the physical significance of the factorisation and its role for quantum statistics. In particular, we study the effect of phase-space interference on the factorisation properties of a superposition state.to appear in J. Phys. A, LaTeX, 13 pages, no figures. More information on http://www.technion.ac.il/~brif/science.htmlQuantum Physicstext