On two superintegrable nonlinear oscillators in N dimensions

Abstract

We consider the classical superintegrable Hamiltonian system given by $H=T+U=p^2/2(1+\lambda q^2)+{\omega }^2{q}^2/2(1+\lambda q^2)$, where U is known to be the "intrinsic" oscillator potential on the Darboux spaces of nonconstant curvature determined by the kinetic energy term T and parametrized by λ. We show that H is Stackel equivalent to the free Euclidean motion, a fact that directly provides a curved Fradkin tensor of constants of motion for H. Furthermore, we analyze in terms of λ the three different underlying manifolds whose geodesic motion is provided by T. As a consequence, we find that H comprises three different nonlinear physical models that, by constructing their radial effective potentials, are shown to be two different nonlinear oscillators and an infinite barrier potential. The quantization of these two oscillators and its connection with spherical confinement models is briefly discussed.

11 pages; based on the contribution to the Manolo Gadella Fest-60 years-in-pucelandia, "Recent advances in time-asymmetric quantum mechanics, quantization and related topics" hold in Valladolid (Spain), 14-16th july 2010

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Cited by in corpus (8)

• Quantum mechanics on spaces of nonconstant curvature: the oscillator problem and superintegrability
• A new exactly solvable quantum model in N dimensions
• Superintegrable Oscillator and Kepler Systems on Spaces of Nonconstant Curvature via the Stäckel Transform
• The classical Taub-Nut System: factorization, spectrum generating algebra and solution to the equations of motion
• Superintegrable deformations of superintegrable systems : Quadratic superintegrability and higher-order superintegrability
• The classical Darboux III oscillator: factorization, Spectrum Generating Algebra and solution to the equations of motion
• Explicit isomorphisms for the symmetry algebras of continuous and discrete isotropic oscillators
• Exact solution of Schrödinger equation for the complex Morse potential to investigate physical systems with position-dependent complex mass