In this paper, we prove the off-shell equation satisfied by the transfer matrix associated with the XXZ spin-12 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.
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Avan et al. (2015) studied this question.
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