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

Abstract: A superintegrable finite model of the quantum isotropic oscillator in two dimensions is introduced. It is defined on a uniform lattice of triangular shape. The constants of the motion for the model form an SU(2) symmetry algebra. It is found that the dynamical difference eigenvalue equation can be written in terms of creation and annihilation operators. The wavefunctions of the Hamiltonian are expressed in terms of two known families of bivariate Krawtchouk polynomials; those of Rahman and those of Tratnik. These polynomials form bases for SU(2) irreducible representations. It is further shown that the pair of eigenvalue equations for each of these families are related to each other by an SU(2) automorphism. A finite model of the anisotropic oscillator that has wavefunctions expressed in terms of the same Rahman polynomials is also introduced. In the continuum limit, when the number of grid points goes to infinity, standard two-dimensional harmonic oscillators are obtained. The analysis provides the $N\rightarrow \infty$ limit of the bivariate Krawtchouk polynomials as a product of one-variable Hermite polynomials.