## Abstract

In this review, we show how advances in the theory of magnetic pseudodifferential operators (magnetic ΨDO) can be put to good use in space-adiabatic perturbation theory (SAPT). As a particular example, we extend results of [24] to a more general class of magnetic fields: we consider a single particle moving in a periodic potential which is subjected to a weak and slowly-varying electromagnetic field. In addition to the semiclassical parameter ε ≪ 1 which quantifies the separation of spatial scales, we explore the influence of an additional parameter λ that allows us to selectively switch off the magnetic field. We find that even in the case of magnetic fields with components in C_{b}^{∞}(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 [24].

Original language | English |
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Pages (from-to) | 233-260 |

Number of pages | 28 |

Journal | Reviews in Mathematical Physics |

Volume | 23 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2011 Apr |

Externally published | Yes |

## Keywords

- Bloch electron
- Magnetic field
- Weyl calculus
- pseudodifferential operators

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics