Space-adiabatic perturbation theory

G. Panati, H. Spohn, S. Teufel

Adv. Theor. Math. Phys. 7 (2003), 145-204.
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Zusammenfassung

We study approximate solutions to the Schrödinger equation iε∂ψ_t(x)/∂t = H(x,-iε∇_x) ψ_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 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.