02-34 Gianluca Panati, Herbert Spohn, Stefan Teufel
Space-adiabatic perturbation theory (181K, Latex2e) Jan 25, 02
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Abstract. We study approximate solutions to the Schroedinger equation $i\epsi\partial\psi_t(x)/\partial t = H(x,-i\epsi\nabla_x)\,\psi_t(x)$ 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 H_f 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 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.

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