Space-Adiabatic Perturbation Theory

We study approximate solutions to the time dependent Schrodinger equation i epsilon partial derivative(t)psi(t)(x)/partial derivative t = H(x, -i epsilon del(x)) psi(t) (x) 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 epsilon. As a consequence, associated to that energy band there exists a subspace of L-2(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.