Generalized quantum Zernike Hamiltonians: Polynomial Higgs-type algebras and algebraic derivation of the spectrum

We consider the quantum analog of the generalized Zernike systems given by the Hamiltonian: $${\widehat{\mathcal{H}}}_N={\widehat{p}}_1^2+{\widehat{p}}_2^2+\sum _{k=1}^N{\gamma }_k({\widehat{q}}_1{\widehat{p}}_1+{\widehat{q}}_2{\widehat{p}}_2{)}^k,$$ with canonical operators ${\widehat{q}}_i,{\widehat{p}}_i$ and arbitrary coefficients ${\gamma }_k$. 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 in two dimensions. 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 $\mathrm{\Phi }$ is shown to factorize into two commuting components $\mathrm{\Phi }={\mathrm{\Phi }}_1{\mathrm{\Phi }}_2$. This framework enables an algebraic determination of the possible energy spectra of the model for the cases $1\le N\le 5$, 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 $N\ge 1$ and any value of the coefficients ${\gamma }_k$. In addition, all of these results can be interpreted as higher-order 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