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

TL;DR

This paper connects scalar products of Bethe states with $q$-Racah polynomials using Leonard triples, providing explicit solutions and determinant formulas for Bethe roots in integrable models.

Contribution

It introduces a novel approach linking Leonard triples to scalar products of Bethe states, deriving explicit formulas and relations for Bethe roots.

Findings

Scalar products expressed as linear combinations of $q$-Racah polynomials.

Explicit solutions for Belliard-Slavnov linear systems.

Determinant formulas for inhomogeneous Bethe roots.

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.