Space-Adiabatic Perturbation Theory

TL;DR

This paper develops a systematic perturbation method for approximating solutions to the Schrödinger equation with Hamiltonians having isolated energy bands, effectively suppressing interband transitions and deriving effective Hamiltonians for intraband dynamics.

Contribution

It introduces a new perturbation scheme for space-adiabatic problems, extending existing methods to compute effective Hamiltonians with suppressed interband transitions.

Findings

Interband transitions are suppressed to any order in epsilon.

Existence of an almost invariant subspace under time evolution.

Development of a systematic scheme for effective Hamiltonians.

Abstract

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.