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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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