# Quantum superintegrable Zernike system

## TL;DR

This paper explores the quantum Zernike system, revealing its superintegrability and separability in multiple coordinate systems, leading to new polynomial solutions and algebraic structures involving P"oschl-Teller potentials.

## Contribution

It introduces a quantum perspective on the Zernike system, uncovers its superintegrability, and derives new polynomial solutions through separation in various coordinate systems.

## Key findings

- The quantum Zernike system separates in three coordinate systems.
- Eigen-solutions involve Legendre, Gegenbauer, and Jacobi polynomials.
- The separation constants form a superintegrable cubic Higgs algebra.

## Abstract

We consider the differential equation that Zernike proposed to classify aberrations of wavefronts in a circular pupil, whose value at the boundary can be nonzero. On this account the quantum Zernike system, where that differential equation is seen as a Schr\"odinger equation with a potential, is special in that it has a potential and boundary condition that are not standard in quantum mechanics. We project the disk on a half-sphere and there we find that, in addition to polar coordinates, this system separates in two additional coordinate systems (non-orthogonal on the pupil disk), which lead to Schr\"odinger-type equations with P\"oschl-Teller potentials, whose eigen-solutions involve Legendre, Gegenbauer and Jacobi polynomials. This provides new expressions for separated polynomial solutions of the original Zernike system that are real. The operators which provide the separation constants are found to participate in a superintegrable cubic Higgs algebra.