Classical and quantum superintegrable systems on the sphere and the hyperbolic 2-space
DOI:
https://doi.org/10.14311/AP.2025.65.0520Keywords:
superintegrable systems, factorisation of Hamiltonians, Tremblay-Turbiner-Winternitz Hamiltonan systemsAbstract
We present two superintegrable Hamiltonian systems in two dimensions, defined on the sphere and on the hyperbolic plane. These systems are generalised à la Tremblay-Turbiner-Winternitz (TTW), involving the introduction of a real parameter k > 0, with the aim of extending superintegrable Hamiltonian systems to curved spaces in a way similar to the TTW system on the plane. We carry out both classical and quantum analyses of these new systems. We prove that the superintegrability of the initial systems (i.e. when k = 1) is preserved when k is rational, as in the TTW case. A detailed study of their classical counterparts and trajectories is also included.
Downloads
References
M. A. del Olmo, M. A. Rodríguez, P. Winternitz. Integrable systems based on SU(p, q) homogeneous manifolds. Journal of Mathematical Physics 34(11):5118– 5139, 1993. https://doi.org/10.1063/1.530346
N. W. Evans. Superintegrability in classical mechanics. Physical Review A 41(10):5666–5676, 1990. https://doi.org/10.1103/PhysRevA.41.5666
J. A. Calzada, M. A. del Olmo, M. A. Rodríguez. Classical superintegrable SO(p, q) Hamiltonian systems. Journal of Geometry and Physics 23(1):14–30, 1997. https://doi.org/10.1016/S0393-0440(96)00043-5
J. A. Calzada, M. A. del Olmo, M. A. Rodríguez. Pseudo-orthogonal groups and integrable dynamical systems in two dimensions. Journal of Mathematical Physics 40(1):188–209, 1999. https://doi.org/10.1063/1.532768
J. A. Calzada, J. Negro, M. A. del Olmo, M. A. Rodríguez. Contraction of superintegrable Hamiltonian systems. Journal of Mathematical Physics 41(1):317–336, 2000. https://doi.org/10.1063/1.533147
J. A. Calzada, J. Negro, M. A. del Olmo. Superintegrable quantum u(3) systems and higher rank factorizations. Journal of Mathematical Physics 47(4):043511, 2006. https://doi.org/10.1063/1.2191360
J. A. Calzada, Ş. Kuru, J. Negro, M. A. d. Olmo. Intertwining symmetry algebras of quantum superintegrable systems on the hyperboloid. Journal of Physics A: Mathematical and Theoretical 41(25):255201, 2008. https://doi.org/10.1088/1751-8113/41/25/255201
J. A. Calzada, Ş. Kuru, J. Negro, M. A. del Olmo. Dynamical algebras of general two-parametric Pöschl–Teller Hamiltonians. Annals of Physics 327(3):808–822, 2012. https://doi.org/10.1016/j.aop.2011.12.014
F. Tremblay, A. V. Turbiner, P. Winternitz. An infinite family of solvable and integrable quantum systems on a plane. Journal of Physics A: Mathematical and Theoretical 42(24):242001, 2009. https://doi.org/10.1088/1751-8113/42/24/242001
F. Tremblay, A. V. Turbiner, P. Winternitz. Periodic orbits for an infinite family of classical superintegrable systems. Journal of Physics A: Mathematical and Theoretical 43(1):015202, 2010. https://doi.org/10.1088/1751-8113/43/1/015202
J. Friš, V. Mandrosov, Y. A. Smorodinsky, et al. On higher symmetries in quantum mechanics. Physics Letters 16(3):354–356, 1965. https://doi.org/10.1016/0031-9163(65)90885-1
P. Winternitz, Y. A. Smorodinsky, M. Uhlíř, J. Friš. Symmetry groups in classical and quantum mechanics. Soviet Journal of Nuclear Physics 4:444–450, 1967.
M. F. Rañada, M. Santander. Superintegrable systems on the two-dimensional sphere S2 and the hyperbolic plane H2. Journal of Mathematical Physics 40(10):5026– 5057, 1999. https://doi.org/10.1063/1.533014
I. Marquette, C. Quesne. New families of superintegrable systems from Hermite and Laguerre exceptional orthogonal polynomials. Journal of Mathematical Physics 54(4):042102, 2013. https://doi.org/10.1063/1.4798807
G. S. Pogosyan, K. B. Wolf, A. Yakhno. Superintegrable classical Zernike system. Journal of Mathematical Physics 58(7):072901, 2017. https://doi.org/10.1063/1.4990793
G. S. Pogosyan, C. Salto-Alegre, K. B. Wolf, A. Yakhno. Quantum superintegrable Zernike system. Journal of Mathematical Physics 58(7):072101, 2017. https://doi.org/10.1063/1.4990794
A. M. Escobar-Ruiz, P. Winternitz, İ. Yurduşen. General Nth-order superintegrable systems separating in polar coordinates. Journal of Physics A: Mathematical and Theoretical 51(40):40LT01, 2018. https://doi.org/10.1088/1751-8121/aadc23
F. Correa, L. Inzunza, I. Marquette. Non-Hermitian superintegrable systems. Journal of Physics A: Mathematical and Theoretical 56(34):345207, 2023. https://doi.org/10.1088/1751-8121/ace506
E. Celeghini, Ş. Kuru, J. Negro, M. A. del Olmo. A unified approach to quantum and classical TTW systems based on factorizations. Annals of Physics 332:27–37, 2012. https://doi.org/10.1016/j.aop.2013.01.008
T. Hakobyan, O. Lechtenfeld, A. Nersessian, et al. Integrable generalizations of oscillator and coulomb systems via action-angle variables. Physics Letters A 376(5):679–686, 2012. https://doi.org/10.1016/j.physleta.2011.12.034
T. Hakobyan, O. Lechtenfeld, A. Nersessian, et al. Action-angle variables and novel superintegrable systems. Physics of Particles and Nuclei 43(5):577–582, 2012. https://doi.org/10.1134/S1063779612050152
M. F. Rañada. The Tremblay-Turbiner-Winternitz system on spherical and hyperbolic spaces: Superintegrability, curvature-dependent formalism and complex factorization. Journal of Physics A: Mathematical and Theoretical 47(16):165203, 2014. https://doi.org/10.1088/1751-8113/47/16/165203
I. Marquette, A. Parr. Superintegrability and deformed oscillator realizations of quantum TTW Hamiltonians on constant-curvature manifolds and with reflections in a plane. Journal of Physics A: Mathematical and Theoretical 57(13):135201, 2024. https://doi.org/10.1088/1751-8121/ad2e3f
L. Infeld, T. E. Hull. The factorization method. Reviews of Modern Physics 23(1):21–68, 1951. https://doi.org/10.1103/RevModPhys.23.21
E. Schrödinger. A method of determining quantum-mechanical eigenvalues and eigenfunctions. In Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences, vol. 46, pp. 9–16. Royal Irish Academy, 1940. http://www.jstor.org/stable/20490744
E. Schrödinger. Further studies on solving eigenvalue problems by factorization. In Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences, vol. 46, pp. 183–206. Royal Irish Academy, 1941. http://www.jstor.org/stable/20490756
E. G. Kalnins, J. M. Kress, W. Miller. Superintegrability and higher order integrals for quantum systems. Journal of Physics A: Mathematical and Theoretical 43(26):265205, 2010. https://doi.org/10.1088/1751-8113/43/26/265205
W. Miller, S. Post, P. Winternitz. Classical and quantum superintegrability with applications. Journal of Physics A: Mathematical and Theoretical 46(42):423001, 2013. https://doi.org/10.1088/1751-8113/46/42/423001
A. Romaniega. Sistemas superintegrables en Mecánica Clásica y en Mecánica Cuántica: Generalización en la ésfera S2 y formulación geométrica de la Mecánica [In Spanish; Superintegrable systems in classical and quantum mechanics: Generalisation on the
S2-sphere and the geometric formulation of mechanics]. Bachelor’s thesis, University of Valladolid, 2016.
G. Pöschl, E. Teller. Bemerkungen zur Quantenmechanik des anharmonischen Oszillators [In German; Remarks on the quantum mechanics of the anharmonic oscillator]. Zeitschrift für Physik 83(3):143–151, 1933. https://doi.org/10.1007/BF01331132
Ş. Kuru, J. Negro. Factorizations of one-dimensional classical systems. Annals of Physics 323(2):413–431, 2008. https://doi.org/10.1016/j.aop.2007.10.004
M. Blazquez, J. Negro. SO(2, 2) representations in polar coordinates and Pöschl-Teller potentials. Journal of Physics A: Mathematical and Theoretical 57(19):195204, 2024. https://doi.org/10.1088/1751-8121/ad3d45
Ş. Kuru, I. Marquette, J. Negro. The general Racah algebra as the symmetry algebra of generic systems on pseudo-spheres. Journal of Physics A: Mathematical and Theoretical 53(40):405203, 2020. https://doi.org/10.1088/1751-8121/abadb7
F. Correa, M. A. del Olmo, I. Marquette, J. Negro. Polynomial algebras from su(3) and a quadratically superintegrable model on the two sphere. Journal of Physics A: Mathematical and Theoretical 54(1):015205, 2020. https://doi.org/10.1088/1751-8121/abc909
J. C. L. Vieyra, A. V. Turbiner. Tremblay-Turbiner-Winternitz (TTW) system at integer index k: Polynomial algebras of integrals. arXiv math-ph:2503.09502, 2025. https://doi.org/10.48550/arXiv.2503.09502
Downloads
Published
Issue
Section
License
Copyright (c) 2025 Mariano A. del Olmo, Álvaro Romaniega
This work is licensed under a Creative Commons Attribution 4.0 International License.
