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


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 E2 and S2 and for a family of systems defined on constant curvature manifolds. The procedure results effective in many cases including Tremblay-Turbiner-Winternitz and three-particle Calogero systems.

Key words: superintegrable Hamiltonian systems; polynomial first integrals.

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References

  1. 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.
  2. 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.
  3. Chanu C., Degiovanni L., Rastelli G., Superintegrable three-body systems on the line, J. Math. Phys. 49 (2008), 112901, 10 pages, arXiv:0802.1353.
  4. Jauch J.M., Hill E.L., On the problem of degeneracy in quantum mechanics, Phys. Rev. 57 (1940), 641-645.
  5. Kalnins E.G., Kress J.M., Pogosyan G.S., Miller Jr. W., Completeness of superintegrability in two-dimensional constant-curvature spaces, J. Phys. A: Math. Gen. 34 (2001), 4705-4720, math-ph/0102006.
  6. 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 20-24, 2012).
  7. 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.
  8. Maciejewski A.J., Przybylska M., Yoshida H., Necessary conditions for classical super-integrability of a certain family of potentials in constant curvature spaces, J. Phys. A: Math. Theor. 43 (2010), 382001, 15 pages, arXiv:1004.3854.
  9. 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.
  10. 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.

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