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

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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