Motion on constant curvature surfaces and a quantization
of some position dependent mass systems

Classical motion on constant curvature surfaces will be analysed and in particular the harmonic oscillator-like system in two dimensions will
be proved to be superintegrable. A quantization of these systems will be
carried out in several alternative ways. This is a report of results of several papers coworked with M.F. Rañada and M. Santander.

References

[1] P.M. Mathews and M. Lakshmanan, "On a unique nonlinear oscillator", Quart. Appl. Math. 32, 215{218 (1974)

[2] J.F. Cariñena, M.F. Rañada, M. Santander and M. Senthilvelan, "A non-linear Oscillator with quasi-Harmonic behaviour: two-and n-dimensional
Oscillators", Nonlinearity 17, 1941{63 (2004)

[3] J.F. Cariñena, M.F. Rañada and M. Santander, "One-dimensional model of a Quantum non-linear Harmonic Oscillator", Rep. Math. Phys.
54, 375{83 (2004)

[4]
J.F. Cariñena, M.F. Rañada and M. Santander, "A quantum exactly solvable nonlinear oscillator with quasi-harmonic behaviour", Ann. Phys. 322, 434{459 (2007)

[5] J.F. Cariñena, M.F. Rañada and M. Santander, "The quantum harmonic oscillator on the sphere and the hyperbolic plane", Ann. Phys. 322, 2249{2278 (2007)

[6] J.F. Cariñena, M.F. Rañada and M. Santander, "The quantum harmonic oscillator on the sphere and the hyperbolic plane: k- dependent formalism, polar coordinates and hypergeometric functions", J. Math. Phys. 48,
102106 (2007)