Invariant eigenoperators and energy gap for some Hamiltonians describing photonic nonlinear interaction

On the basis of the Heisenberg equation of motion and Schrodinger's correspondence , we search for so-called invariant eigenoperators for some Hamiltonians describing photonic nonlinear interactions. In this way the energy-level gap of the Hamiltonians can be naturally obtained. The characteristic polynomial theory has been fully employed in our derivation.