Canonical transformations to phase variables in quantum oscillator systems. A group theoretic solution

We consider the problem of finding the unitary transformation which interwines a self-adjoint quantum Hamiltonian operator 1/2 p2 + V(Q) in L2-type function spaces with a local weight function, to its phase-variable form - id/d zeta, self-adjoint in a second Hilbert space. This can be done in the context of the Heisenberg-Weyl algebra only for the free fall case. Within the sl(2,R) algebra one may consider potentials V(Q) = #betta#Q2 + #betta#Q-2 for all real #betta#, #betta# corresponding to harmonic or repulsive oscillators, or a free particle, with a centrifugal barrier or centripetal well. This treatment exhausts the class of Hamiltonians for which the problem of quantum canonical transformations to phase variables may be solved entirely within the framework of group theory. The results naturally agree with the ones obtained from the Mello-Moshinsky equations, except that the spectra are here automatically matched through the (in general nonlocal) measure in the second Hilbert space. The present treatment does not require, thus, the introduction of the Moshinsky-Seligman ambiguity group. The algebra sl(2,R) replaces the quantum phase space associated to the Heisenberg-Weyl algebra. Furthermore, we examine in detail a range of singular potentials - those which include a strong centripetal well - where the Hamiltonian has a one-parameter family of self-adjoint extensions, and where the discrete spectra are neither unique nor lower bound. Finally, we find a set of new generating relations for Whittaker functions. (author)