局域缺陷周期结构中缺陷模的均匀化描述
Homogenized description of defect modes in periodic structures with localized defects
摘要：
具有周期系数的能量守恒波动方程的空间局部化初始条件的振幅随着时间的推移而分散（空间扩展）和衰减。这种色散与基础微分算子的连续谱和离散本征值的缺乏有关。在周期性介质中引入空间局部扰动导致缺陷模式，即能量保持被捕获和空间局部化的状态。本文研究一维周期薛定谔算子的弱局部扰动。这种扰动会产生这种缺陷模式，并且与来自连续谱的离散本征值的出现相关。由于这些孤立的本征值位于附近的光谱带的边缘，有很强的尺度分离之间的介质周期和缺陷模式的本地化长度。因此，束缚态具有多尺度结构：一个“载波布洛赫波”乘以一个“波包络”，这是由一个均匀化的薛定谔算子与相关的有效质量，取决于光谱带边，这是分叉的网站。我们的分析是基于在布洛赫准动量空间的本征值问题的重新制定，使用Gelfand-Bloch变换和Lyapunov-Schmidt减少到一个封闭的方程的近带边频率分量的束缚态。后一个方程的重标度产生波包络的均匀化有效方程，以及分叉本征值和本征函数的近似。
英文摘要：
A spatially localized initial condition for an energy-conserving wave equation with periodic coefficients disperses (spatially spreads) and decays in amplitude as time advances. This dispersion is associated with the continuous spectrum of the underlying differential operator and the absence of discrete eigenvalues. The introduction of spatially localized perturbations in a periodic medium leads to defect modes, states in which energy remains trapped and spatially localized. In this paper we study weak, localized perturbations of one-dimensional periodic Schrödinger operators. Such perturbations give rise to such defect modes, and are associated with the emergence of discrete eigenvalues from the continuous spectrum. Since these isolated eigenvalues are located near a spectral band edge, there is strong scale-separation between the medium period and the localization length of the defect mode. Bound states therefore have a multi-scale structure: a "carrier Bloch wave" times a "wave envelope", which is governed by a homogenized Schrödinger operator with associated effective mass, depending on the spectral band edge which is the site of the bifurcation. Our analysis is based on a reformulation of the eigenvalue problem in Bloch quasi-momentum space, using the Gelfand-Bloch transform and a Lyapunov-Schmidt reduction to a closed equation for the near-band-edge frequency components of the bound state. A rescaling of the latter equation yields the homogenized effective equation for the wave envelope, and approximations to bifurcating eigenvalues and eigenfunctions.
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