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A superintegrable finite model of the quantum isotropic oscillator in twodimensions is introduced. It is defined on a uniform lattice of triangularshape. The constants of the motion for the model form an SU(2) symmetryalgebra. It is found that the dynamical difference eigenvalue equation can bewritten in terms of creation and annihilation operators. The wavefunctions ofthe Hamiltonian are expressed in terms of two known families of bivariateKrawtchouk polynomials; those of Rahman and those of Tratnik. These polynomialsform bases for SU(2) irreducible representations. It is further shown that thepair of eigenvalue equations for each of these families are related to eachother by an SU(2) automorphism. A finite model of the anisotropic oscillatorthat has wavefunctions expressed in terms of the same Rahman polynomials isalso introduced. In the continuum limit, when the number of grid points goes toinfinity, standard two-dimensional harmonic oscillators are obtained. Theanalysis provides the $N\rightarrow \infty$ limit of the bivariate Krawtchoukpolynomials as a product of one-variable Hermite polynomials.

A superintegrable finite oscillator in two dimensions with SU(2) symmetry