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

SIGMA 3 (2007), 025, 10 pages      math-ph/0702048
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

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.

pdf (215 kb)   ps (154 kb)   tex (14 kb)


  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.

Previous article   Next article   Contents of Volume 3 (2007)