SIGMA 5 (2009), 008, 24 pages      arXiv:0901.3081      https://doi.org/10.3842/SIGMA.2009.008

Structure Theory for Second Order 2D Superintegrable Systems with 1-Parameter Potentials

Ernest G. Kalnins ^a, Jonathan M. Kress ^b, Willard Miller Jr. ^c and Sarah Post ^c
a) Department of Mathematics, University of Waikato, Hamilton, New Zealand
^b) School of Mathematics, The University of New South Wales, Sydney NSW 2052, Australia
^c) School of Mathematics, University of Minnesota, Minneapolis, Minnesota, 55455, USA

Received November 26, 2008, in final form January 14, 2009; Published online January 20, 2009

Abstract
The structure theory for the quadratic algebra generated by first and second order constants of the motion for 2D second order superintegrable systems with nondegenerate (3-parameter) and or 2-parameter potentials is well understood, but the results for the strictly 1-parameter case have been incomplete. Here we work out this structure theory and prove that the quadratic algebra generated by first and second order constants of the motion for systems with 4 second order constants of the motion must close at order three with the functional relationship between the 4 generators of order four. We also show that every 1-parameter superintegrable system is Stäckel equivalent to a system on a constant curvature space.

Key words: superintegrability; quadratic algebras.

pdf (305 kb)   ps (195 kb)   tex (25 kb)

References

1. Kalnins E.G., Kress J.M., Miller W. Jr., Second order superintegrable systems in conformally flat spaces. I. 2D classical structure theory, J. Math. Phys. 46 (2005), 053509, 28 pages.
2. Kalnins E.G., Kress J.M., Miller W. Jr., Second order superintegrable systems in conformally flat spaces. II. The classical 2D Stäckel transform, J. Math. Phys. 46 (2005), 053510, 15 pages.
3. Daskaloyannis C., Ypsilantis K., Unified treatment and classification of superintegrable systems with integrals quadratic in momenta on a two-dimensional manifold, J. Math. Phys. 47 (2006), 042904, 38 pages, math-ph/0412055.
4. Kalnins E.G., Kress J.M., Miller W. Jr., Pogosyan G.S., Completeness of superintegrability in two-dimensional constant curvature spaces, J. Phys. A: Math. Gen. 34 (2001), 4705-4720, math-ph/0102006.
5. Kalnins E.G., Kress J.M., Miller W. Jr., Second order superintegrable systems in conformally flat spaces. V. 2D and 3D quantum systems, J. Math. Phys. 47 (2006), 093501, 25 pages.
6. Wojciechowski S., Superintegrability of the Calogero-Moser system, Phys. Lett. A 95 (1983), 279-281.
7. Evans N.W., Superintegrability in classical mechanics, Phys. Rev. A 41 1990, 5666-5676.
   Evans N.W., Group theory of the Smorodinsky-Winternitz system, J. Math. Phys. 32 (1991), 3369-3375.
8. Evans N.W., Super-integrability of the Winternitz system, Phys. Lett. A 147 (1990), 483-486.
9. Fris J., Mandrosov V., Smorodinsky Ya.A., Uhlír M., Winternitz P., On higher symmetries in quantum mechanics, Phys. Lett. 16 (1965), 354-356.
10. Fris J., Smorodinskii Ya.A., Uhlír M., Winternitz P., Symmetry groups in classical and quantum mechanics, Soviet J. Nuclear Phys. 4 (1967), 444-450.
11. Bonatos D., Daskaloyannis C., Kokkotas K., Deformed oscillator algebras for two-dimensional quantum superintegrable systems, Phys. Rev. A 50 (1994), 3700-3709, hep-th/9309088.
12. Daskaloyannis C., Quadratic Poisson algebras of two-dimensional classical superintegrable systems and quadratic associative algebras of quantum superintegrable systems, J. Math. Phys. 42 (2001), 1100-1119, math-ph/0003017.
13. Grosche C., Pogosyan G.S., Sissakian A.N., Path integral discussion for Smorodinsky-Winternitz potentials. I. Two- and three-dimensional Euclidean space, Fortschr. Phys. 43 (1995), 453-521, hep-th/9402121.
14. Kalnins E.G., Kress J.M., Winternitz P., Superintegrability in a two-dimensional space of non-constant curvature, J. Math. Phys. 43 (2002), 970-983, math-ph/0108015.
15. Kalnins E.G., Miller W. Jr., Post S., Models of quadratic quantum algebras and their relation to classical superintegrable systems, submitted.
16. Kalnins E.G., Miller W. Jr., Pogosyan G.S., Superintegrability and associated polynomial solutions. Euclidean space and the sphere in two dimensions, J. Math. Phys. 37 (1996) 6439-6467.
17. Kalnins E. G., Miller W. Jr., Post S., Kalnins E.G., Miller W. Jr., Post S., Wilson polynomials and the generic superintegrable system on the 2-sphere, J. Phys. A: Math. Theor. 40 (2007), 11525-11538.
18. Kalnins E.G., Miller W. Jr., Post S., Quantum and classical models for quadratic algebras associated with second order superintegrable systems, SIGMA 4 (2008), 008, 21 pages, arXiv:0801.2848.
19. Koenigs G., Sur les géodésiques a intégrales quadratiques, A note appearing in "Lecons sur la théorie générale des surfaces", G. Darboux, Vol. 4, Chelsea Publishing, 1972, 368-404.
20. Hietarinta J., Grammaticos B., Dorizzi B., Ramani A., Coupling-constant metamorphosis and duality between integrable Hamiltonian systems, Phys. Rev. Lett. 53 (1984), 1707-1710.
21. Boyer C.P., Kalnins E.G., Miller W. Jr., Stäckel-equivalent integrable Hamiltonian systems, SIAM J. Math. Anal. 17 (1986), 778-797.
22. Kalnins E.G., Kress J.M., Miller W. Jr., Winternitz P., Superintegrable systems in Darboux spaces, J. Math. Phys. 44 (2003), 5811-5848, math-ph/0307039.
23. Tsiganov A., Addition theorems and the Drach superintegrable systems, J. Phys. A: Math. Theor. 41 (2008), 335204, 16 pages, arXiv:0805.3443.
24. Tsiganov A., Leonard Euler: addition theorems and superintegrable systems, arXiv:0810.1100.
25. Kalnins E.G., Kress J.M., Miller W. Jr., Fine structure for 3D second order superintegrable systems: 3-parameter potentials, J. Phys. A: Math. Theor. 40 (2007), 5875-5892.
26. Verrier P.E., Evans N.W., A new superintegrable Hamiltonian, J. Math. Phys. 49 (2008), 022902, 8 pages, arXiv:0712.3677.
27. Evans N.W., Verrier P.E., Superintegrability of the caged anisotropic oscillator, J. Math. Phys. 49 (2008), 092902, 10 pages, arXiv:0808.2146.