Modified algebraic Bethe ansatz for XXZ chain on the segment - III - Proof

TL;DR

This paper proves the off-shell equation for the transfer matrix of the XXZ spin-1/2 chain on a segment with generic boundaries, revealing the structure of the modified algebraic Bethe ansatz.

Contribution

It provides a rigorous proof of the off-shell equation and the structure of the modified creation operator in the algebraic Bethe ansatz for the XXZ chain with boundaries.

Findings

Off-shell equation for the transfer matrix is established.

Modified creation operator acts with an inhomogeneous term.

Eigenvalues and Bethe equations include inhomogeneous contributions.

Abstract

In this paper, we prove the off-shell equation satisfied by the transfer matrix associated with the XXZ spin-$\frac12$ chain on the segment with two generic integrable boundaries acting on the Bethe vector. The essential step is to prove that the expression of the action of a modified creation operator on the Bethe vector has an off-shell structure which results in an inhomogeneous term in the eigenvalues and Bethe equations of the corresponding transfer matrix.