Quantum Super-Integrable Systems as Exactly Solvable Models

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.

This is a contribution to the Vadim Kuznetsov Memorial Issue on Integrable Systems and Related Topics, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/

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• Completeness of superintegrability in two-dimensional constant curvature spaces
• Quadratic Poisson algebras for two dimensional classical superintegrable systems and quadratic associative algebras for quantum superintegrable systems
• Superintegrability on the two dimensional hyperboloid II

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• First Integrals from Conformal Symmetries: Darboux-Koenigs Metrics and Beyond
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• New infinite families of $N$th-order superintegrable systems separating in Cartesian coordinates
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• Poisson Algebras and 3D Superintegrable Hamiltonian Systems
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• Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras
• Adding Potentials to Superintegrable Systems with Symmetry
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• An infinite family of Dunkl type superintegrable curved Hamiltonians through coalgebra symmetry: Oscillator and Kepler-Coulomb models
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• A Kaluza-Klein Reduction of Super-integrable Systems
• Examples of Complete Solvability of 2D Classical Superintegrable Systems
• Laplace equations, conformal superintegrability and Bôcher contractions
• Completion of the integrable coupling systems
• Remarks on the geometric quantization of a class of harmonic oscillator type potentials