APPLICATIONS OF MAGNETIC ΨDO TECHNIQUES TO SAPT
APPLICATIONS OF MAGNETIC ΨDO TECHNIQUES TO SAPT
复制标题
磁ΨDO技术在SAPT中的应用
DOI：
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发表时间：
2011
期刊：
影响因子：
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通讯作者：
Max Lein
中科院分区：
文献类型：
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作者：
G. Nittis;Max Lein
在这篇综述中，我们展示了如何磁赝微分算子（磁赝微分算子）理论的进展，可以很好地利用空间绝热微扰理论（SAPT）。作为一个特殊的例子，我们将[24]的结果推广到更一般的磁场类别：我们考虑在周期势中运动的单个粒子，该势受到弱且缓慢变化的电磁场的影响。除了量化空间尺度分离的半经典参数ε 1之外，我们还探索了一个附加参数λ的影响，该参数允许我们选择性地关闭磁场。我们发现，即使在磁场的情况下，例如，对于恒定磁场，Panati，Spohn和Teufel的结果成立，即对于每一个孤立的Bloch带族，存在一个与之相关联的几乎不变子空间L2（λ d）和一个有效的哈密顿量，该哈密顿量在这个几乎不变子空间内产生动力学。在孤立的非简并布洛赫带的情况下，完整的量子动力学可以近似为与[24]中发现的半经典运动方程相关联的哈密顿流。
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 e ≪ 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 , 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 L2(ℝd) 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].