Space-adiabatic perturbation theory

We study approximate solutions to the time-dependent Schr̈odinger equation with the Hamiltonian given as the Weyl quantization of the symbol H(q,p) taking values in the space of bounded operators on the Hilbert space H_f of fast "internal" degrees of freedom. By assumption H(q,p) has an isolated energy band. Using a method of Nenciu and Sordoni [NeSo] we prove that interband transitions are suppressed to any order in ε. As a consequence, associated to that energy band there exists a subspace of L²(R^d,H_f) almost invariant under the unitary time evolution. We develop a systematic perturbation scheme for the computation of effective Hamiltonians which govern approximately the intraband time evolution. As examples for the general perturbation scheme we discuss the Dirac and Born-Oppenheimer type Hamiltonians and we reconsider also the time-adiabatic theory.