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 hyperbolic spaces, (κ > 0) and (κ < 0). The curvature κ is considered as a parameter and all the results are formulated in explicit dependence on κ. 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 Mr and Nϕ). This technique leads to a proof of the superintegrability of the Tremblay–Turbiner–Winternitz system on (κ > 0) and (κ < 0), and to the explicit expression of the constants of motion.