磁PsiDO技术在空间绝热微扰理论中的应用
Applications of Magnetic PsiDO Techniques to Space-adiabatic Perturbation Theory
摘要：
在这篇综述中，我们展示了如何磁赝微分算子理论（磁$Ψ$ DO）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[PST 03]的结果推广到更一般的一类磁场：我们考虑单个粒子在周期势中运动，该势受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$\eps \ll 1$之外，我们还探索了其他参数的影响，这些参数允许我们选择性地关闭磁场。我们发现，即使在磁场分量为$C_b^{\infty}(\R^d)$的情况下，e. G.对于恒定磁场，Panati，Spohn和Teufel的结果成立，即对于每个孤立的Bloch带族，存在一个与之相关联的几乎不变子空间$L^2(\R^d)$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[PST 03]中发现的半经典运动方程相关的哈密顿流。
英文摘要：
In this review, we show how advances in the theory of magnetic pseudodifferential operators (magnetic $Ψ$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].
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