Finite two-dimensional oscillator: II. The radial model

A finite two-dimensional radial oscillator of (N + 1)(2) points is proposed, with the dynamical Lie algebra so(4) = su(2)(x) circle plus su(2)(y) examined in part I of this work, but reduced by a subalgebra chain so(4) superset of so(3) superset of so(2). As before, there are a finite number of energies and angular momenta; the Casimir spectrum of the new chain provides the integer radii 0 less than or equal to rho less than or equal to N, and the 2 rho + 1 discrete angles on each circle rho are obtained from the finite Fourier transform of angular momenta. The wavefunctions of the finite radial oscillator are so (3) Clebsch-Gordan coefficients. We define here the Hankel-Hahn transforms (with dual Hahn polynomials) as finite-N unitary approximations to Hankel integral transforms (with Bessel functions), obtained in the contraction limit N --> infinity.