Applications of Magnetic PsiDO Techniques to Space-adiabatic Perturbation Theory

TL;DR

This paper reviews how magnetic pseudodifferential operator theory enhances space-adiabatic perturbation theory, extending results to more general magnetic fields and demonstrating effective Hamiltonians for quantum dynamics in periodic potentials.

Contribution

It extends space-adiabatic perturbation theory to broader magnetic field classes, providing new effective Hamiltonians and invariant subspaces for quantum systems with magnetic fields.

Findings

Effective Hamiltonians exist for each isolated Bloch band.

Quantum dynamics can be approximated by semiclassical equations.

Results hold even for magnetic fields with smooth bounded components.

Abstract

In this review, we show how advances in the theory of magnetic pseudodifferential operators (magnetic $\Psi$DO) can be put to good use in space-adiabatic perturbation theory (SAPT). As a particular example, we extend results of [PST03] to a more general class of magnetic fields: we consider a single particle moving in a periodic potential which is subjectd to a weak and slowly-varying electromagnetic field. In addition to the semiclassical parameter $\eps \ll 1$ which quantifies the separation of spatial scales, we explore the influence of additional parameters that allow us to selectively switch off the magnetic field. We find that even in the case of magnetic fields with components in $C_b^{\infty}(\R^d)$, e. g. for constant magnetic fields, the results of Panati, Spohn and Teufel hold, i.e. to each isolated family of Bloch bands, there exists an associated almost invariant subspace of $L^2(\R^d)$ and an effective hamiltonian which generates the dynamics within this almost invariant subspace. In case of an isolated non-degenerate Bloch band, the full quantum dynamics can be approximated by the hamiltonian flow associated to the semiclassical equations of motion found in [PST03].