TY - JOUR
T1 - Space-adiabatic perturbation theory
JF - Adv. Theor. Math. Phys. 7 (2003) 145-204
Y1 - 2003
A1 - Gianluca Panati
A1 - Herbert Spohn
A1 - Stefan Teufel
AB - 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.
PB - International Press
UR - http://hdl.handle.net/1963/3041
U1 - 1292
U2 - Mathematics
U3 - Mathematical Physics
ER -