2025/02/04 by Campoamor-Stursberg, Rutwig, Herranz, Francisco J., Latini, Danilo +2 · 1 citation
#22E60 #37J35 #46N50 #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Physics (quant-ph)
paper · doi:10.48550/arxiv.2502.02491
We consider the quantum analog of the generalized Zernike systems given by the Hamiltonian: H N =p12+p22+∑k=1N γk (q1 p1+q2 p2)k , with canonical operators qi, pi and arbitrary coefficients γk. 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 h2. 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=Φ1 Φ2. 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≥ 2 and any value of the coefficients γk, 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.