Generalized quantum anharmonic oscillator using an operator ordering approach

We construct a generalized expression for the normal ordering of ( a + a † ) m for integral values of m and use the result to study the quantum anharmonic oscillator problem in the Heisenberg approach. In particular, we derive generalized expressions for energy eigenvalues and frequency shifts for the Hamiltonian H = (( x 2 )/2) + ((ẋ 2 )/2) + (λ/ m ) x m . We also derive a closed-form first-order multiscale perturbation theoretic operator solution of this Hamiltonian with a view to generalize some recent results of Bender and Bettencourt (1996 Phys. Rev. Lett. 77 4114, 1996 Phys. Rev. D 54 1710).