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: $$\hat{\mathcal{H}}_N =\hat{p}_1^2+\hat{p}_2^2+\sum_{k=1}^N γ_k (\hat{q}_1 \hat{p}_1+\hat{q}_2 \hat{p}_2)^k ,$$ 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 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 $Φ$ 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 $1\le N \le 5$, 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 $γ_k$. 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