 0234 Gianluca Panati, Herbert Spohn, Stefan Teufel
 Spaceadiabatic perturbation theory
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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 BornOppenheimer
type Hamiltonians and we reconsider also the timeadiabatic theory.
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