周期结构的振动和局域扰动与缺陷模的分岔
Oscillatory and localized perturbations of periodic structures and the bifurcation of defect modes
摘要：
令$Q(x)$表示实数线上的周期函数。薛定谔算子$H_Q=-\partial_x^2+Q(x)$的$L^2(\mathbb{R})-$谱等于由开放谱隙分隔的闭合实区间的并集。在这篇文章中，我们研究了离散特征值（点谱）的分歧到频谱间隙的运营商$H_{Q+q_ε}$，其中$q_ε$是空间本地化和高度振荡的意义上，它的傅立叶变换，$\widehat{q}_ε$是集中在高频率。我们的假设意味着，$q_ε$可能是逐点大，但$q_ε$是小的平均意义。对于特殊的情况下，其中$q_ε(x)=q(x,x/ε)$与$q(x,y)$光滑，实值，本地化的$x$，周期或几乎周期的$y$，分叉的特征值是在距离$ε^4$阶的频谱间隙的下边缘。我们得到了分支特征值和特征函数的首阶渐近性。这个分叉的基础是与$(b_*)^{\rm th}$谱带的下边缘相关的有效哈密顿量：$H^ε_{\rm eff}=-\partial_x A_{b_*,\rm eff}\partial_x - ε^2 B_{b_*,\rm eff} \times δ(x)$，其中$δ(x)$是狄拉克分布，有效介质参数$A_{b_*,\rm eff},B_{b_*,\rm eff}>0$是明确的，与$ε$无关。我们考虑的潜力是一个自然的模型，波在介质中的传播与本地化，高对比度和快速波动的材料参数的背景周期性介质。
英文摘要：
Let $Q(x)$ denote a periodic function on the real line. The Schrödinger operator, $H_Q=-\partial_x^2+Q(x)$, has $L^2(\mathbb{R})-$ spectrum equal to the union of closed real intervals separated by open spectral gaps. In this article we study the bifurcation of discrete eigenvalues (point spectrum) into the spectral gaps for the operator $H_{Q+q_ε}$, where $q_ε$ is spatially localized and highly oscillatory in the sense that its Fourier transform, $\widehat{q}_ε$ is concentrated at high frequencies. Our assumptions imply that $q_ε$ may be pointwise large but $q_ε$ is small in an average sense. For the special case where $q_ε(x)=q(x,x/ε)$ with $q(x,y)$ smooth, real-valued, localized in $x$, and periodic or almost periodic in $y$, the bifurcating eigenvalues are at a distance of order $ε^4$ from the lower edge of the spectral gap. We obtain the leading order asymptotics of the bifurcating eigenvalues and eigenfunctions. Underlying this bifurcation is an effective Hamiltonian associated with the lower edge of the $(b_*)^{\rm th}$ spectral band: $H^ε_{\rm eff}=-\partial_x A_{b_*,\rm eff}\partial_x - ε^2 B_{b_*,\rm eff} \times δ(x)$ where $δ(x)$ is the Dirac distribution, and effective-medium parameters $A_{b_*,\rm eff},B_{b_*,\rm eff}>0$ are explicit and independent of $ε$. The potentials we consider are a natural model for wave propagation in a medium with localized, high-contrast and rapid fluctuations in material parameters about a background periodic medium.
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