Defect Modes for Dislocated Periodic Media
Defect Modes for Dislocated Periodic Media
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DOI：
10.1007/s00220-020-03787-0
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发表时间：
2020-06
影响因子：
2.4
通讯作者：
A. Drouot;C. Fefferman;M. Weinstein
中科院分区：
文献类型：
--
作者：
A. Drouot;C. Fefferman;M. Weinstein
本文研究了一维周期介质中受振幅绝热位错扰动的缺陷模。如果周期背景允许狄拉克点--色散曲线的线性交叉--那么错位算子在其本质谱中获得间隙。对于该模型（及其蜂窝状类似物），Favorman等人（Proc Natl Acad Sci USA 111（24）：8759-8763，2014，Mem Am Math Soc 247（1173）：118，2017，Ann PDE 2（2）：80，2016，2D Mater 3：1，2016）构建了能量在差距内的缺陷模式（以前导顺序）。这些分叉从一个有效的狄拉克算子的本征模。在这里，我们解决以下开放的问题：Doalldefect模式出现从狄拉克运营商本征模的分叉？这些模式是否允许扩展到所有的顺序？我们对这两个问题都作出了积极的答复。我们的方法依赖于（a）整体算子的预解估计;（B）高振荡势的散射理论[Dr 18 a，Dr 18 B，Dr 18 c]。它导致了连续位错系统中缺陷态的拓扑稳定性的理解-与体边缘对应[Dr 18 d]。
We study defect modes in a one-dimensional periodic medium perturbed by an adiabatic dislocation of amplitude. If the periodic background admits a Dirac point—a linear crossing of dispersion curves—then the dislocated operator acquires a gap in its essential spectrum. For this model (and its honeycomb analog) Fefferman et al. (Proc Natl Acad Sci USA 111(24):8759–8763, 2014, Mem Am Math Soc 247(1173):118, 2017, Ann PDE 2(2):80, 2016, 2D Mater 3:1, 2016) constructed (at leading order in) defect modes with energies within the gap. These bifurcate from the eigenmodes of an effective Dirac operator. Here we address the following open problems:Doalldefect modes arise as bifurcations from the Dirac operator eigenmodes?Do these modes admit expansions to all order in?We respond positively to both questions. Our approach relies on (a) resolvent estimates for the bulk operators; (b) scattering theory for highly oscillatory potentials [Dr18a, Dr18b, Dr18c]. It has led to an understanding of the topological stability of defect states in continuous dislocated systems—in connection with the bulk-edge correspondence [Dr18d].