Determine completeness of the proposed basis for the H_min eigenspace at energy (M−1)^2+1

Determine whether the set of states |φ^{(2,k)}_{N,M} defined in equation (\ref{eq: second eigenspace Hmin}) forms a complete basis for the H_{min} eigenspace with eigenvalue (M−1)^2+1 (equation (\ref{eq: energy second Hmin block})) in block (N,M), and provide a proof or counting argument establishing completeness or characterize any additional states needed.

Background

For the top eigenspace of H_min, the authors construct a complete basis and fully diagonalize H, revealing square-integer spectra. In the next eigenspace at energy (M−1)^2+1, they propose a family of states |φ^{(2,k)}_{N,M} and show H acts tridiagonally within the subspace they span.

However, they lack a counting proof that these states exhaust the entire H_min eigenspace at that energy. Establishing completeness (or identifying necessary additional states) would extend the analytic diagonalization to this and potentially further lower eigenspaces.