Analytic proof of completeness and solution-count equality for the two Bethe Ansatz equation sets

Prove analytically that the total number of physical solutions of the Bethe Ansatz equations for the invariant subspaces G_M^+ and G_M^- equals the respective dimensions d_+=\sum_{k=0}^{M}\binom{N}{k} and d_-=\sum_{k=0}^{N-1-M}\binom{N}{k}, thereby establishing that these two Bethe Ansatz equation sets constitute the complete spectrum of the open XXZ chain with constrained non-diagonal boundary fields.

Background

For the open spin-1/2 XXZ chain with constrained non-diagonal boundary fields, the paper introduces two conventional Bethe Ansatz equation systems associated with invariant subspaces G_M+ and G_M-. Their dimensions are d_+=\sum_{k=0}{M}\binom{N}{k} and d_-=\sum_{k=0}{N-1-M}\binom{N}{k}=2N-d_+.

Numerical calculations indicate that the number of physical solution sets of each Bethe Ansatz equation system matches the dimension of its corresponding invariant subspace. If established analytically, this would prove completeness of the two Bethe Ansatz descriptions and confirm that their combined solutions produce the full transfer-matrix and Hamiltonian spectrum. The paper explicitly leaves this analytic justification unresolved.

References

Numerical results indicate that the total number of physical solutions of the BAEs equals the dimension of each subspace, and that these two sets of BAEs constitute the complete spectrum of the transfer matrix and hence of the Hamiltonian. However, an analytic proof of this observation remains an open question.

— Bethe-root configurations and spectral degeneracy in the open XXZ chain with degenerate boundaries  (2608.20815 - Wu et al., 21 Aug 2026) in Section 2, subsection “Bethe Ansatz equations”