标题： 部分周期薛定谔算符的方向弹道输运 显示英文标题

标题： Directional Ballistic Transport for Partially Periodic Schrödinger Operators

摘要： 我们研究了定义在$\mathbb{R}^d$和$\mathbb{Z}^d$上的薛定谔算子的输运性质，其势函数在某些方向上是周期性的，在其他方向上则具有紧支集。这类系统已知会产生“表面态”，这些态被限制在势函数支撑附近的区域。我们证明了，在非常温和的假设下，一类表面态表现出所谓的方向性弹道输运，即在周期方向上具有强形式的弹道输运而在其他方向上不存在弹道输运。此外，在$\mathbb{Z}^2$上，我们表明一大类表面态表现出方向性弹道输运。在附录中，我们推广了 Simon 关于纯点态不存在弹道输运的经典结果 [36]，并证明了一个关于散射态弹道输运的民间定理。特别是，这个最后的结果允许我们证明周期条带模型中$\ell^2(\mathbb{Z}^2)$的一大类子集的弹道输运。 显示英文摘要

摘要： We study the transport properties of Schr\"{o}dinger operators on $\mathbb{R}^d$ and $\mathbb{Z}^d$ with potentials that are periodic in some directions and compactly supported in the others. Such systems are known to produce "surface states" that are confined near the support of the potential. We show that, under very mild assumptions, a class of surface states exhibits what we describe as directional ballistic transport, consisting of a strong form of ballistic transport in the periodic directions and its absence in the other directions. Furthermore, on $\mathbb{Z}^2$, we show that a dense set of surface states exhibit directional ballistic transport. In appendices, we generalize Simon's classic result on the absence of ballistic transport for pure point states [36], and prove a folklore theorem on ballistic transport for scattering states. In particular, this final result allows for a proof of ballistic transport for a dense set subset of $\ell^2(\mathbb{Z}^2)$ for periodic strip models.