Superintegrable quantum u(3)--systems and higher rank factorizations

Abstract

A class of two-dimensional superintegrable systems on a constant curvature surface is considered as the natural generalization of some well known one-dimensional factorized systems. By using standard methods to find the shape-invariant intertwining operators we arrive at a $so(6)$ dynamical algebra and its Hamiltonian hierarchies. We pay attention to those associated to certain unitary irreducible representations that can be displayed by means of three-dimensional polyhedral lattices. We also discuss the role of superpotentials in this new context.

20 pages, 4 figures

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• Algebraic (super-)integrability from commutants of subalgebras in universal enveloping algebras
• Polynomial algebras from $su(3)$ and the generic model on the two sphere
• The general Racah algebra as the symmetry algebra of generic systems on pseudo--spheres
• Applications of the potential algebras of the two-dimensional Dirac-like operators
• Groups, Jacobi functions and rigged Hilbert spaces
• Demkov-Fradkin tensor for curved harmonic oscillators
• Revisiting the Symmetries of the Quantum Smorodinsky-Winternitz System in D Dimensions
• Algebraic structures and Hamiltonians from the equivalence classes of 2D conformal algebras
• Non-Hermitian superintegrable systems