Generalized classical and quantum Zernike Hamiltonians
Francisco J. Herranz, Alfonso Blasco, Rutwig Campoamor-Stursberg, Ivan Gutierrez-Sagredo, Danilo Latini, Ian Marquette
TL;DR
The paper generalizes the Zernike Hamiltonians to a family $H_N$ that adds higher-order momentum-dependent terms, yielding classical and quantum systems with superintegrability. It proves that the classical system possesses a complete polynomial (Higgs-type) symmetry algebra of order $2N-1$ and develops a quantum algebraic framework via a deformed $\mathfrak{sl}(2,\mathbb{R})$ and a deformed oscillator to determine spectra; for $N=4$ it provides explicit integrals, the algebraic structure, and two energy-series $E_I(n)$ and $E_{II}(n)$, with real spectra under suitable complex parameter choices. The work situates the generalized Zernike family as a bridge between curved oscillator perturbations and higher-order superintegrability, offering tractable algebraic structures and solvable spectra for low $N$ and curved-space interpretations (spherical, hyperbolic, Euclidean).
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.