Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 3 (2007), 030, 23 pages      math-ph/0702084      https://doi.org/10.3842/SIGMA.2007.030
Contribution to the Proceedings of the Coimbra Workshop on Geometric Aspects of Integrable Systems

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

José F. Cariñena a, Manuel F. Rañada a and Mariano Santander b
a) Departamento de Física Teórica, Facultad de Ciencias Universidad de Zaragoza, 50009 Zaragoza, Spain
b) Departamento de Física Teórica, Facultad de Ciencias Universidad de Valladolid, 47011 Valladolid, Spain

Received October 31, 2006, in final form January 24, 2007; Published online February 24, 2007

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.

Key words: deformed oscillator; integrability, super-integrability; Hamilton-Jacobi separability; Hamilton-Jacobi super-separability; quantum solvable systems.

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