
SIGMA 8 (2012), 070, 12 pages arXiv:1210.3126
https://doi.org/10.3842/SIGMA.2012.070
Contribution to the Special Issue “Superintegrability, Exact Solvability, and Special Functions”
Superintegrable Extensions of Superintegrable Systems
Claudia M. Chanu ^{a}, Luca Degiovanni ^{b} and Giovanni Rastelli ^{c}
^{a)} Dipartimento di Matematica, Università di Torino, Torino, via Carlo Alberto 10, Italy
^{b)} Formerly at Dipartimento di Matematica, Università di Torino, Torino, via Carlo Alberto 10, Italy
^{c)} Independent researcher, cna Ortolano 7, Ronsecco, Italy
Received July 30, 2012, in final form September 27, 2012; Published online October 11, 2012
Abstract
A procedure to extend a superintegrable system into a new superintegrable one is systematically tested for the known systems on E^{2} and S^{2} and for a family of systems defined on constant curvature manifolds. The procedure results effective in many cases including TremblayTurbinerWinternitz and threeparticle Calogero systems.
Key words:
superintegrable Hamiltonian systems; polynomial first integrals.
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References
 Chanu C., Degiovanni L., Rastelli G., First integrals of extended
Hamiltonians in n+1 dimensions generated by powers of an operator,
SIGMA 7 (2011), 038, 12 pages, arXiv:1101.5975.
 Chanu C., Degiovanni L., Rastelli G., Generalizations of a method for
constructing first integrals of a class of natural Hamiltonians and some
remarks about quantization, J. Phys. Conf. Ser. 343 (2012),
012101, 15 pages, arXiv:1111.0030.
 Chanu C., Degiovanni L., Rastelli G., Superintegrable threebody systems on the
line, J. Math. Phys. 49 (2008), 112901, 10 pages,
arXiv:0802.1353.
 Jauch J.M., Hill E.L., On the problem of degeneracy in quantum mechanics,
Phys. Rev. 57 (1940), 641645.
 Kalnins E.G., Kress J.M., Pogosyan G.S., Miller Jr. W., Completeness of
superintegrability in twodimensional constantcurvature spaces,
J. Phys. A: Math. Gen. 34 (2001), 47054720,
mathph/0102006.
 Kalnins E.G., Kress J.M., Miller Jr. W., Talk given by J. Kress during the
conference "Superintegrability, Exact Solvability, and Special Functions"
(Cuernavaca, February 2024, 2012).
 Kalnins E.G., Kress J.M., Miller Jr. W., Tools for verifying classical and
quantum superintegrability, SIGMA 6 (2010), 066, 23 pages,
arXiv:1006.0864.
 Maciejewski A.J., Przybylska M., Yoshida H., Necessary conditions for classical
superintegrability of a certain family of potentials in constant curvature
spaces, J. Phys. A: Math. Theor. 43 (2010), 382001,
15 pages, arXiv:1004.3854.
 Rodríguez M.A., Tempesta P., Winternitz P., Reduction of superintegrable
systems: the anisotropic harmonic oscillator, Phys. Rev. E
78 (2008), 046608, 6 pages, arXiv:0807.1047.
 Tremblay F., Turbiner A.V., Winternitz P., An infinite family of solvable and
integrable quantum systems on a plane, J. Phys. A: Math. Theor.
42 (2009), 242001, 10 pages, arXiv:0904.0738.

