Invariant 'eigen-operator' of the square of Schroedinger operator for deriving energy-level gap

We propose the conception of invariant 'eigen-operator' of the square of the Schroedinger operator, which can be used to derive energy-level gap formulas for some dynamic Hamiltonians. We list some examples, coupled oscillators models and the degenerate amplifier, etc., to demonstrate the feasibility of this approach