Discretization of superintegrable systems on a plane

摘要

We construct difference analogues of so called Smorodinsky-Winternitz superinte-grable systems in the Euclidean plane. Using methods of umbral calculus, we obtain difference equations for generalized isotropic harmonic oscillator on the uniform lattice, and also its solution in terms of power series. In the case of gauge-rotated Hamiltonian, the solution is a polynomial, well-defined in the whole plane.