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

TL;DR

This paper explores the algebraic structure of the radial harmonic oscillator using the J-matrix method, revealing that Zernike polynomials naturally arise in the coherent state expansions and encode solutions for a charged particle in a magnetic field on the Poincaré disc.

Contribution

It introduces a novel connection between the tridiagonal matrix representation of the RHO and Zernike polynomials, linking algebraic methods with special functions and quantum solutions.

Findings

Zernike polynomials appear as expansion coefficients in coherent states

The tridiagonal structure encodes bound state solutions of a 2D Schrödinger operator

The approach relates algebraic matrix methods to physical quantum systems

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.