Space-Adiabatic Perturbation Theory

We study approximate solutions to the Schrödinger equation iε∂ψt(x)/∂t = H(x,−iε∇x)ψt(x) with the Hamiltonian H the Weyl quantization of the symbol H(q, p) taking values in the space of bounded operators on the Hilbert space Hf of fast “internal” degrees of freedom. By assumption H(q, p) has an isolated energy band. 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 L(R,Hf) 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.