Generalized quantum Zernike Hamiltonians: polynomial higgs-type algebras and algebraic derivation of the spectrum ^*

Abstract We consider the quantum analogue of the generalized Zernike systems given by the Hamiltonian: H ^ N = p ^ 1 2 + p ^ 2 2 + ∑ k = 1 N γ k ( q ^ 1 p ^ 1 + q ^ 2 p ^ 2 ) k , with canonical operators q ^ i , 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 ⩽ N ⩽ 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 ⩾ 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 analysed and applied to the isotropic oscillator on the sphere, hyperbolic and Euclidean spaces.