A Super-Integrable Two-Dimensional Non-Linear Oscillator with an Exactly Solvable Quantum Analog

Abstract

Two super-integrable and super-separable classical systems which can be considered as deformations of the harmonic oscillator and the Smorodinsky-Winternitz in two dimensions are studied and identified with motions in spaces of constant curvature, the deformation parameter being related with the curvature. In this sense these systems are to be considered as a harmonic oscillator and a Smorodinsky-Winternitz system in such bi-dimensional spaces of constant curvature. The quantization of the first system will be carried out and it is shown that it is super-solvable in the sense that the Schrödinger equation reduces, in three different coordinate systems, to two separate equations involving only one degree of freedom.

This is a contribution to the Proc. of workshop on Geometric Aspects of Integrable Systems (July 17-19, 2006; Coimbra, Portugal), published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/

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Cited by in corpus (7)

• Quasi-Lie schemes: theory and applications
• n-dimensional PDM non-linear oscillators: Linearizability and Euler-Lagrange or Newtonian invariance
• An update on the classical and quantum harmonic oscillators on the sphere and the hyperbolic plane in polar coordinates
• On the symmetries of a nonlinear non-polynomial oscillator
• $n$-dimensional PDM-damped harmonic oscillators: Linearizability, and exact solvability
• Revisiting the Symmetries of the Quantum Smorodinsky-Winternitz System in D Dimensions
• Algebraic derivation of the Energy Eigenvalues for the quantum oscillator defined on the Sphere and the Hyperbolic plane