Time dependence of operators in anharmonic quantum oscillators: Explicit perturbative analysis

An explicit, order-by-order perturbative solution, valid over extended time scales, for the time dependence of operators of anharmonic oscillators, is developed within the framework of the method of normal forms. The freedom of choice of the zeroth-order term and, concurrently in the higher-order corrections, is exploited to develop a minimal normal form (MNF). The expansion for the eigenvalues of the perturbed Hamiltonian in a standard form is independent of the choice. However, the simple form obtained for the time dependence of the perturbative solution is more suitable than any other choice for application to high-lying excited states, as it offers a renormalized form for the propagator.