The Superintegrable Zernike System

We present a resume of this year’s work on what we call the Zernike system. It stems from a differential equation proposed by Frits Zernike in 1934 to describe wavefronts at circular optical pupils through a basis of polynomial solutions on the unit disk and free boundary conditions. This system entails a classical model and a quantum model. The classical model leads to closed elliptic orbits while the quantum model yields bases of polynomial wavefunctions that separate in a manifold of coordinate systems, where only the polar one is orthogonal. Special functions that appear in the solutions and interbasis expansions include the Legendre, Gegenbauer, Jacobi, Hahn and Racah polynomials, as well as special Clebsch–Gordan and 6j coefficients. The underlying symmetry is a cubic Higgs superintegrable algebra.