On discrete WKB methods for resonance electromagnetic traps

We consider a model quantum Hamiltonian of a charge in a resonance electromagnetic trap. Using the operator averaging method, we obtain an effective quantum operator that asymptotically describes the anharmonic part of the Hamiltonian. We show that the operator becomes a second-order difference operator in a specially chosen quantum action-angle representation. Using the discrete WKB method for this difference equation, we obtain the semiclassical WKB asymptotics of the spectrum and stationary states of the charge.