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Bulgarian Journal of Physics
Print: 1310-0157 Online: 1314-2666
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RECOGNIZED BY THE EUROPEAN PHYSICAL SOCIETY 
Bulg. J. Phys. vol.52 no.s1 (2025), pp. 139-145
Generalized Classical and Quantum Zernike Hamiltonians
F.J. Herranz1, A. Blasco1, R. Campoamor-Stursberg2, I. Gutierrez-Sagredo3, D. Latini4, I. Marquette5
1Departamento de Fìsica, Universidad de Burgos, 09001 Burgos, Spain
2Instituto de Matemàtica Interdisciplinar & Departamento de Àlgebra, Geo\-metrìa y Topologìa, UCM, 28040 Madrid, Spain
3Departamento de Matemàticas y Computaciòn, Universidad de Burgos, 09001 Burgos, Spain
4Dipartimento di Matematica "Federigo Enriques", Università degli Studi di Milano & INFN Sezione di Milano, 20133 Milano, Italy
5Department of Mathematical and Physical Sciences, La Trobe University, Bendigo, VIC 3552, Australia
go back1Departamento de Fìsica, Universidad de Burgos, 09001 Burgos, Spain
2Instituto de Matemàtica Interdisciplinar & Departamento de Àlgebra, Geo\-metrìa y Topologìa, UCM, 28040 Madrid, Spain
3Departamento de Matemàticas y Computaciòn, Universidad de Burgos, 09001 Burgos, Spain
4Dipartimento di Matematica "Federigo Enriques", Università degli Studi di Milano & INFN Sezione di Milano, 20133 Milano, Italy
5Department of Mathematical and Physical Sciences, La Trobe University, Bendigo, VIC 3552, Australia
Abstract. A superintegrable generalization of the classical and quantum Zernike systems is reviewed. The corresponding Hamiltonians are endowed with higher-order integrals and can be interpreted as higher-order superintegrable perturbations of the 2D spherical (Higgs), hyperbolic, and Euclidean harmonic oscillators. As a new result, the complete polynomial Higgs-type symmetry algebra of the generalized classical system is presented. For the generalized quantum system, the symmetry algebra and the spectra are provided for a representative case.