Quantum averaging for driven systems with resonances

We discuss the effects of resonances in driven quantum systems within the c ontext of quantum averaging techniques in the Floquet representation We con sider in particular iterative methods of KAM type and the extensions needed to take into account resonances. The approach consists in separating the c oupling terms into resonant and nonresonant components at a given scale of time and intensity. The nonresonant part can be treated with perturbative t echniques, which we formulate in terms of KAM-type unitary transformations that are close to the identity. These can be interpreted as avenging proced ures with respect to the dynamics defined by effective uncoupled Hamiltonia ns. The resonant parts are treated with a different kind of unitary transfo mations that are not close to the identity, and are adapted to the structur e of the resonances. They can be interpreted as a renormalization of the un coupled Hamiltonian, that yields an effective dressed Hamiltonian, around w hich a perturbation expansion can be developed. The combination of these tw o ingredients provides a strongly improved approximation technique. (C) 200 0 Elsevier Science B.V. All rights reserved.