SIGMA 3 (2007), 025, 10 pages      math-ph/0702048      https://doi.org/10.3842/SIGMA.2007.025
Contribution to the Vadim Kuznetsov Memorial Issue

Quantum Super-Integrable Systems as Exactly Solvable Models

Allan P. Fordy
Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK

Received November 14, 2006, in final form February 05, 2007; Published online February 14, 2007

Abstract
We consider some examples of quantum super-integrable systems and the associated nonlinear extensions of Lie algebras. The intimate relationship between super-integrability and exact solvability is illustrated. Eigenfunctions are constructed through the action of the commuting operators. Finite dimensional representations of the quadratic algebras are thus constructed in a way analogous to that of the highest weight representations of Lie algebras.

Key words: quantum integrability; super-integrability; exact solvability; Laplace-Beltrami.

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References

1. Boyer C.P., Kalnins E.G., Winternitz P., Completely integrable relativistic Hamiltonian systems and separation of variables in Hermitian hyperbolic spaces, J. Math. Phys. 24 (1983), 2022-2034.
2. 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.
3. Fordy A.P., Symmetries, ladder operators and quantum integrable systems, Glasg. Math. J. 47 (2005), 65-75.
4. Fordy A.P., Darboux related quantum integrable systems on a constant curvature surface, J. Geom. Phys. 56 (2006), 1709-1727.
5. Gilmore R., Lie groups, Lie algebras and some of their applications, Wiley, New York, 1974.
6. Harnad J., Vinet L., Yermolayeva O., Zhedanov A., Two-dimensional Krall-Sheffer polynomials and integrable systems, J. Phys. A: Math. Gen. 34 (2001), 10619-10625.
7. Humphreys J.E., Introduction to Lie algebras and representation theory, Springer-Verlag, Berlin, 1972.
8. Infeld L., Hull T., The factorization method, Rev. Modern Phys. 23 (1951), 21-68.
9. Kalnins E.G., Kress J.M., Pogosyan G.S., Miller W.Jr., Completeness of superintegrability in two-dimensional constant-curvature spaces, J. Phys. A: Math. Gen. 34 (2001), 4705-4720, math-ph/0102006.
10. Kalnins E.G., Miller W.Jr., Hakobyan Ye.M., Pogosyan G.S., Superintegrability on the two-dimensional hyperboloid. II, J. Math. Phys. 40 (1999), 2291-2306, quant-ph/9907037.
11. Kalnins E.G., Miller W.Jr., Pogosyan G.S., Exact and quasiexact solvability of second-order superintegrable quantum systems. I. Euclidean space preliminaries, J. Math. Phys. 47 (2006), 033502, 30 pages, math-ph/0412035.
12. Krall H.L., Sheffer I.M., Orthogonal polynomials in two variables, Ann. Mat. Pura Appl. (4) 76 (1967), 325-376.
13. Kuznetsov V.B., Hidden symmetry of the quantum Calogero-Moser system, Phys. Lett. A 218 (1996), 212-222, solv-int/9509001.
14. Tempesta P., Turbiner A.V., Winternitz P., Exact solvability of superintegrable systems, J. Math. Phys. 42 (2001), 4248-4257, hep-th/0011209.
15. Vinet L., Zhedanov A., Two-dimensional Krall-Sheffer polynomials and quantum systems on spaces of constant curvature, Lett. Math. Phys. 65 (2003), 83-94.