Zernike polynomials from the tridiagonalization of the radial harmonic oscillator in displaced Fock states

Abstract

We revisit the J-matrix method for the one dimensional radial harmonic oscillator (RHO) and construct its tridiagonal matrix representation within an orthonormal basis phi(z)n of L2 (R+);parametrized by a fixed z in the complex unit disc D and n = 0,1,2,.... Remarkably, for fixed n,and varying z in D, the system phi(z)n forms a family of Perelomov-type coherent states associated with the RHO. For each fixed n, the expansion of phi(z)n over the basis (fs) of eigenfunctions of the RHO yields coefficients cn,s(z; z) precisely given by two-dimensional complex Zernike polynomials. The key insight is that the algebraic tridiagonal structure of RHO contains the complete information about the bound state solutions of the two-dimensional Schr\"odinger operator describing a charged particle in a magnetic field (of strength proportional to B > 1/2) on the Poincar\'e disc D.