The Tremblay-Turbiner-Winternitz system on spherical and hyperbolic spaces : Superintegrability, curvature-dependent formalism and complex factorization

Abstract

The higher-order superintegrability of the Tremblay-Turbiner-Winternitz system (related to the harmonic oscillator) is studied on the two-dimensional spherical and hiperbolic spaces, $S_\k^2$ ($\k>0$), and $H_{\k}^2$ ($\k<0$). The curvature $\kappa$ is considered as a parameter and all the results are formulated in explicit dependence of $\kappa$. The idea is that the additional constant of motion can be factorized as the product of powers of two particular rather simple complex functions (here denoted by $M_r$ and $N_{\varphi }$). This technique leads to a proof of the superintegrability of the Tremblay-Turbiner-Winternitz system on $S_\k^2$ ($\k>0$) and $H_{\k}^2$ ($\k<0$), and to the explicit expression of the constants of motion.

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Cited by in corpus (17)

• Superintegrable systems on 3-dimensional curved spaces: Eisenhart formalism and separability
• The Tremblay-Turbiner-Winternitz system as extended Hamiltonian
• Symmetries in superintegrable deformations of oscillator/Coulomb systems: "holomorphic factorization"
• Superintegrable deformations of superintegrable systems : Quadratic superintegrability and higher-order superintegrability
• Lobachevsky geometry in TTW and PW systems
• Extended Hamiltonians, Coupling-Constant Metamorphosis and the Post-Winternitz System
• New superintegrable models on spaces of constant curvature
• Extended Hamiltonians and shift, ladder functions and operators
• Quasi-Bi-Hamiltonian Structures of the 2-Dimensional Kepler Problem
• Kähler geometry for $su(1,N|M)$-superconformal mechanics
• Curvature as an integrable deformation
• Noncompact ${\mathbf{CP}}^N$ as a phase space of superintegrable systems
• Superintegrable dynamics on $H^2$ generated by coupling the Morse and Rosen-Morse potentials
• Superintegrable deformations of oscillator and Coulomb systems
• Complex functions and geometric structures associated to the superintegrable Kepler-related family of systems endowed with generalized Runge-Lenz integrals of motion
• Superintegrability of the Post-Winternitz system on spherical and hyperbolic spaces
• Superintegrability on the 3-dimensional spaces with curvature. Oscillator-related and Kepler-related systems on the Sphere $S^3$ and on the Hyperbolic space $H^3$