Space-Adiabatic Perturbation Theory

2002/01/31 by Gianluca Panati, Herbert Spohn, Stefan Teufel · 2 citations
Physics and Astronomy · Mathematics · #math-ph #math.MP #msc:81Q05 #msc:81Q15

paper · pdf

published as Adv. Theor. Math. Phys. 7 (2003) 145-204 · 49 pages

arxiv created 2003/12/24 · arxiv updated 2009/11/30

Abstract

We study approximate solutions to the Schrödinger equation i\epsi∂ψ_t(x)/∂ t = H(x,-i\epsi∇_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 \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 \citeNS 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 L²(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.

Cited by

• The generalized adiabatic theorem for extended lattice systems
• On the time-dependent Born-Oppenheimer Approximation

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