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⊕su(2)y examined in part I of this work, but reduced by a subalgebra chain so(4)⊃so(3)⊃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⩽ρ⩽N, and the 2ρ + 1 discrete angles on each circle ρ 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→∞.