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

Abstract

We consider the quantum analog of the generalized Zernike systems given by the Hamiltonian: $${\hat{\mathcal{H}}}_N={\hat{p}}_1^2+{\hat{p}}_2^2+\sum _{k=1}^N{\gamma }_k({\hat{q}}_1{\hat{p}}_1+{\hat{q}}_2{\hat{p}}_2{)}^k,$$ with canonical operators ${\hat{q}}_i,{\hat{p}}_i$ and arbitrary coefficients ${\gamma }_k$. 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 $\Phi$ is shown to factorize into two commuting components $\Phi ={\Phi }_1{\Phi }_2$. 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

30 pages, 5 figures. The results have been clarified and two new appendices have been included. New comments and references have been added

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