Generalized quantum Zernike Hamiltonians: Polynomial Higgs-type algebras and algebraic derivation of the spectrum
Rutwig Campoamor-Stursberg, Francisco J. Herranz, Danilo Latini, Ian Marquette, Alfonso Blasco
TL;DR
This work introduces a generalized quantum Zernike framework with Hamiltonians $\hat{\mathcal{H}}_N = \hat{p}_1^2+\hat{p}_2^2+\sum_{k=1}^N \gamma_k (\hat{q}_1 \hat{p}_1+\hat{q}_2 \hat{p}_2)^k$, showing that quantum superintegrability persists for all $N$. The authors construct quantum symmetry algebras—polynomial Higgs-type algebras—that are mapped, via a basis change, to deformed oscillator algebras with a factorized structure function $\Phi=\Phi_1\Phi_2$, enabling an algebraic derivation of spectra. They solve explicitly the quadratic ($N=2$), cubic ($N=3$), and quartic ($N=4$) cases, and prove conjectures for $N=5$, providing two spectral branches and interpretations as superintegrable perturbations of the Zernike system and of curved isotropic oscillators on $\mathbf S^2$, $\mathbf H^2$, and $\mathbf E^2$. The framework paves the way for general $N$ by presenting a precise conjectural structure for the symmetry algebra and spectrum, with potential applications to momentum-dependent perturbations in curved spaces and beyond.
Abstract
We consider the quantum analog of the generalized Zernike systems given by the Hamiltonian: $$ \hat{\mathcal{H}} _N =\hat{p}_1^2+\hat{p}_2^2+\sum_{k=1}^N γ_k (\hat{q}_1 \hat{p}_1+\hat{q}_2 \hat{p}_2)^k , $$ with canonical operators $\hat{q}_i,\, \hat{p}_i$ and arbitrary coefficients $γ_k$. This two-dimensional quantum model, besides the conservation of the angular momentum, exhibits higher-order integrals of motion within the enveloping algebra of the Heisenberg algebra $\mathfrak h_2$. By constructing suitable combinations of these integrals, we uncover a polynomial Higgs-type symmetry algebra that, through an appropriate change of basis, gives rise to a deformed oscillator algebra. The associated structure function $Φ$ is shown to factorize into two commuting components $Φ=Φ_1 Φ_2$. This framework enables an algebraic determination of the possible energy spectra of the model for the cases $N=2,3,4$, the case $N=1$ being canonically equivalent to the harmonic oscillator. Based on these findings, we propose two conjectures which generalize the results for all $N\ge 2$ and any value of the coefficients $γ_k$, that they are explicitly proven for $N=5$. In addition, all of these results can be interpreted as superintegrable perturbations of the original quantum Zernike system corresponding to $N=2$ which are also analyzed and applied to the isotropic oscillator on the sphere, hyperbolic and Euclidean spaces.