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A bispectral q-hypergeometric basis for a class of quantum integrable models

Abstract : For the class of quantum integrable models generated from the q−Onsager algebra, a basis of bispectral multivariable q−orthogonal polynomials is exhibited. In a first part, it is shown that the multivariable Askey-Wilson polynomials with N variables and N+3 parameters introduced by Gasper and Rahman [GR1] generate a family of infinite dimensional irreducible modules for the q−Onsager algebra, whose fundamental generators are realized in terms of the multivariable q−difference and difference operators proposed by Iliev [Il]. Raising and lowering operators extending those of Sahi [Sa2] are also constructed. In a second part, finite dimensional irreducible modules are constructed and studied for a certain class of parameters and if the N variables belong to a discrete support. In this case, the bispectral property finds a natural interpretation within the framework of tridiagonal pairs. In a third part, eigenfunctions of the q−Dolan-Grady hierarchy are considered in the polynomial basis. In particular, invariant subspaces are identified for certain conditions generalizing Nepomechie's relations. In a fourth part, the analysis is extended to the special case q=1. This framework provides a q−hypergeometric formulation of quantum integrable models such as the open XXZ spin chain with generic integrable boundary conditions (q≠1).
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Contributor : Pascal Baseilhac Connect in order to contact the contributor
Submitted on : Friday, June 26, 2015 - 10:08:46 AM
Last modification on : Tuesday, January 11, 2022 - 5:56:09 PM

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  • HAL Id : hal-01168537, version 1
  • ARXIV : 1506.06902



Pascal Baseilhac, Xavier Martin. A bispectral q-hypergeometric basis for a class of quantum integrable models. 2015. ⟨hal-01168537⟩



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