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Space-Adiabatic Perturbation Theory | | Gianluca Panati
; Herbert Spohn
; Stefan Teufel
; | | Date: |
25 Dec 2001 | | Journal: | Adv. Theor. Math. Phys. 7 (2003) 145-204 | | Subject: | Mathematical Physics MSC-class: 81Q05, 81Q15 | math-ph math.MP | | Abstract: | We study approximate solutions to the Schrödinger equation $iepsipartialpsi_t(x)/partial t = H(x,-iepsi
abla_x) psi_t(x)$ 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 $Hi_{
m f}$ 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 $L^2(mathbb{R}^d,Hi _{
m f})$ 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. | | Source: | arXiv, math-ph/0201055 | | Services: | Forum | Review | PDF | Favorites | |
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