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arXiv · 2501.10310

The $q$-Racah polynomials from scalar products of Bethe states II

Pascal Baseilhac·Rodrigo A. Pimenta
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Abstract

The theory of Leonard triples is applied to the derivation of normalized scalar products of on-shell and off-shell Bethe states generated from a Leonard pair. The scalar products take the form of linear combinations of $q$-Racah polynomials with coefficients depending on the off-shell parameters. Upon specializations, explicit solutions for the corresponding Belliard-Slavnov linear systems are obtained. It implies the existence of a determinant formula in terms of inhomogeneous Bethe roots for the $q$-Racah polynomials. Also, a set of relations that determines solutions (Bethe roots) of the corresponding Bethe equations of inhomogeneous type in terms of solutions of Bethe equations of homogenous type is obtained.

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Pascal Baseilhac, Rodrigo A. Pimenta. 2025-01-17. The $q$-Racah polynomials from scalar products of Bethe states II. https://arxiv.org/abs/2501.10310

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