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February 2003 Space-adiabatic perturbation theory
Gianluca Panti, Herbert Spohn, Stefan Teufel
Adv. Theor. Math. Phys. 7(1): 145-204 (February 2003).


We study approximate solutions to the time-dependent Schrodinger equation $i\epsi\partial_t\psi_t(x)/\partial t = H(x,-i\epsi\nabla_x)\,\psi_t(x)$ 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 $\Hi _{\rm f}$ of fast ''internal'' degrees of freedom. By assumption $H(q,p)$ has an isolated energy band. Using a method of Nenciu and Sordoni \cite{NS} we prove that interband transitions are suppressed to any order in $\epsi$. As a consequence, associated to that energy band there exists a subspace of $L^2(\mathbb{R}^d,\Hi _{\rm f})$ 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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Gianluca Panti. Herbert Spohn. Stefan Teufel. "Space-adiabatic perturbation theory." Adv. Theor. Math. Phys. 7 (1) 145 - 204, February 2003.


Published: February 2003
First available in Project Euclid: 4 April 2005

MathSciNet: MR2014961

Rights: Copyright © 2003 International Press of Boston

Vol.7 • No. 1 • February 2003
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