Space-Adiabatic Perturbation Theory

Details Space-Adiabatic Perturbation Theory Space-Adiabatic Perturbation Theory

2002-01-25

arxiv.org/abs/math-ph/0201055v3

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

Physics

Mathematical Physics

49 pages

Scientific paper

We study approximate solutions to the Schr\"odinger equation $i\epsi\partial\psi_t(x)/\partial t = H(x,-i\epsi\nabla_x) \psi_t(x)$ 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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