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arXiv · 1608.04590

The generic quantum superintegrable system on the sphere and Racah operators

Plamen Iliev
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Abstract

We consider the generic quantum superintegrable system on the $d$-sphere with potential $V(y)=\sum_{k=1}^{d+1}\frac{b_k}{y_k^2}$, where $b_k$ are parameters. Appropriately normalized, the symmetry operators for the Hamiltonian define a representation of the Kohno-Drinfeld Lie algebra on the space of polynomials orthogonal with respect to the Dirichlet distribution. The Gaudin subalgebras generated by Jucys-Murphy elements are diagonalized by families of Jacobi polynomials in $d$ variables on the simplex. We define a set of generators for the symmetry algebra and we prove that their action on the Jacobi polynomials is represented by the multivariable Racah operators introduced in arXiv:0705.1469. The constructions also yield a new Lie-theoretic interpretation of the bispectral property for Tratnik's multivariable Racah polynomials.

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Plamen Iliev. 2017-06-12. The generic quantum superintegrable system on the sphere and Racah operators. https://doi.org/10.1007/s11005-017-0978-3

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