Generalized classical and quantum Zernike Hamiltonians

TL;DR

This paper reviews a superintegrable generalization of classical and quantum Zernike systems, revealing higher-order symmetries and spectra, and extending the understanding of superintegrable perturbations of fundamental oscillators.

Contribution

It introduces a new complete polynomial Higgs-type symmetry algebra for the classical system and analyzes the symmetry algebra and spectra for the quantum case.

Findings

Complete polynomial Higgs-type symmetry algebra derived for the classical system.

Symmetry algebra and spectra characterized for the quantum generalized system.

Generalization encompasses higher-order superintegrable perturbations of key oscillators.

Abstract

A superintegrable generalization of the classical and quantum Zernike systems is reviewed. The corresponding Hamiltonians are endowed with higher-order integrals and can be interpreted as higher-order superintegrable perturbations of the 2D spherical (Higgs), hyperbolic, and Euclidean harmonic oscillators. As a new result, the complete polynomial Higgs-type symmetry algebra of the generalized classical system is presented. For the generalized quantum system, the symmetry algebra and the spectra are provided for a representative case.