Space-adiabatic perturbation theory

Abstract

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 Hf 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(Rd,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. © 2003 International Press.