On Bethe vectors in $\mathfrak{gl}_3$-invariant integrable models

TL;DR

This paper introduces a new method for constructing Bethe vectors in $rak{gl}_3$-invariant integrable models, proving their semi-on-shell nature and their on-shell condition when Bethe equations are satisfied.

Contribution

It presents a novel approach to construct on-shell Bethe vectors in $rak{gl}_3$-invariant models and proves their semi-on-shell properties for arbitrary parameters.

Findings

Constructed Bethe vectors are semi-on-shell for any parameters.

Bethe vectors become on-shell when Bethe equations are satisfied.

The method provides a new way to analyze $rak{gl}_3$-invariant integrable systems.

Abstract

We consider quantum integrable models solvable by the nested algebraic Bethe ansatz and possessing $\mathfrak{gl}_3$-invariant $R$-matrix. We study a new recently proposed approach to construct on-shell Bethe vectors of these models. We prove that the vectors constructed by this method are semi-on-shell Bethe vectors for arbitrary values of Bethe parameters. They thus do become on-shell vectors provided the system of Bethe equations is fulfilled.