Unitarity criterion for partitioned Bose–Hubbard Hamiltonians

Let the $(M,L)$-partitioned Hamiltonians be the Hamiltonians referred to as

, and let the corresponding $(M,L)$-partitioned analogue of the fundamental matrix referred to as

have a spectrum. In the leading-order approximation, the partitioned-Hamiltonian unitarity conjecture. Quantum evolution controlled by the $(M,L)$-partitioned Hamiltonians will be unitary provided only that the spectrum of the corresponding $(M,L)$-partitioned analogue of the fundamental matrix is real and non-degenerate. This extends the expected applicability of the preceding result to models with block-off-diagonal perturbations; the statement is presented without proof, and its resolution is not supplied in the source.