We study defect modes in a one-dimensional periodic medium perturbed by an adiabatic dislocation of amplitude $$\delta \ll 1$$ δ ≪ 1 . 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 $$\delta $$ δ ) defect modes with energies within the gap. These bifurcate from the eigenmodes of an effective Dirac operator. Here we address the following open problems: Do all defect modes arise as bifurcations from the Dirac operator eigenmodes? Do these modes admit expansions to all order in $$\delta $$ δ ? 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 ].

中文翻译：

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错位周期性介质的缺陷模式

我们研究了一维周期性介质中的缺陷模式，该介质受到振幅 $$\delta \ll 1$$ δ ≪ 1 的绝热位错的干扰。如果周期性背景允许一个狄拉克点——色散曲线的线性交叉点——那么错位算子在其基本光谱中获得了一个间隙。对于这个模型（及其蜂窝模拟），Fefferman 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） （在 $$\delta $$ δ 中的领先顺序）能量在间隙内的缺陷模式。这些从有效狄拉克算子的本征模式分叉出来。在这里，我们解决以下开放性问题：是否所有的缺陷模式都是从 Dirac 算子本征模式的分岔中产生的？这些模式是否允许对 $$\delta $$ δ 中的所有阶进行扩展？我们对这两个问题都做出了积极的回应。我们的方法依赖于 (a) 对散货运营商的解决方案估计；(b) 高振荡电位的散射理论 [Dr18a、Dr18b、Dr18c]。它导致了对连续错位系统中缺陷态拓扑稳定性的理解——与体边缘对应 [Dr18d] 相关。