## Abstract

We consider a model of an electron in a crystal moving under the influence of an external electric field:Schrödinger’s equation in one spatial dimension with a potential which is the sum of a periodic function V and a smooth function W. We assume that the period of V is much shorter than the scale of variation of W and denote the ratio of these scales by ϵ. We consider the dynamics of semiclassical wavepacket asymptotic (in the limit ϵ↓ 0) solutions which are spectrally localized near to a crossing of two Bloch band dispersion functions of the periodic operator -12∂z2+V(z). We show that the dynamics is qualitatively different from the case where bands are well-separated: at the time the wavepacket is incident on the band crossing, a second wavepacket is ‘excited’ which has opposite group velocity to the incident wavepacket. We then show that our result is consistent with the solution of a ‘Landau–Zener’-type model.

Original language | English (US) |
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Pages (from-to) | 655-698 |

Number of pages | 44 |

Journal | Communications in Mathematical Physics |

Volume | 363 |

Issue number | 2 |

DOIs | |

State | Published - Oct 1 2018 |

Externally published | Yes |

### Bibliographical note

Funding Information:The authors wish to thank George Hagedorn, Jianfeng Lu, and Christof Sparber for stimulating discussions. This research was supported in part by National Science Foundation Grant Nos. DMS-1412560, DMS-1620418 and Simons Foundation Math + X Investigator Award #376319 (Michael I. Weinstein).

Funding Information:

Acknowledgements. The authors wish to thank George Hagedorn, Jianfeng Lu, and Christof Sparber for stimulating discussions. This research was supported in part by National Science Foundation Grant Nos. DMS-1412560, DMS-1620418 and Simons Foundation Math + X Investigator Award #376319 (Michael I. Weinstein).

Publisher Copyright:

© 2018, Springer-Verlag GmbH Germany, part of Springer Nature.