Canonical transforms. III. Configuration and phase descriptions of quantum systems possessing an s l (2,R) dynamical algebra

The purpose of this article is to present a detailed analysis on the quantum mechnical level of the canonical transformation between coordinate-momentum and number-phase descriptions for systems possessing an sl (2,R) dynamical algebra, specifically, the radial harmonic oscillator and pseudo-Coulomb systems. The former one includes the attractive and repulsive oscillators and the free particle, each with an additional ’’centrifugal’’ force, while the latter includes the bound, free and threshold states with an added ’’centrifugal’’ force. This is implemented as a unitary mapping—canonical transform—between the usual Hilbert space ℒ2 of quantum mechanics and a new set of Hilbert spaces on the circle whose coordinate has the meaning of a phase variable. Moreover, the UIR’s D+k of the universal covering group of SL (2,R) realized on the former space are mapped unitarily onto the latter.