发表机构
School of Mathematics and Statistics, Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University（东北师范大学数学与统计学院，数学与交叉科学中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对高对比度弹性亚波长谐振器簇，建立极点扣除极限吸收原理，分解截断预解式并分析共振特性，揭示暗-亮共振混合规律。
AI 中文摘要
我们针对固定的N个互不相交的三维高对比度弹性谐振器簇，在材料对比度趋于无穷大且频率接近亚波长尺度零阈值（ω=δ^(1/2)τ）的联合 regime 中，建立了极点扣除极限吸收原理。通过弹性Dirichlet-to-Neumann映射消除均匀外部区域，并在包含域上应用变分Grushin-Feshbach约化，将截断预解式精确分解为一致有界的正则部分和由以下式子控制的有限秩共振部分：M_δ^±(ω)=δK−ω²I_m∓iiδωΓ₀+O(δ²+δω²)。此外，截断预解式范数与1+∥(M_δ^±(ω))⁻¹∥一致等价，因此所有一致性损失均由有限维通道承载。尽管刚性空间维数为6N，但弹性光学恒等式表明，主导辐射矩阵由单个进入C³的总力映射生成；因此对任意N，Γ₀的秩为3，Γ₀的核维数为6N−3。在力暗共振分支上，第二个非负辐射形式给出阈值费米黄金规则。亮极点宽度为O(δ)，实轴峰值为O(δ^(-3/2))，而二阶亮暗极点宽度为O(δ²)，峰值为O(δ^(-5/2))。该退化理论捕获多个静态本征空间内的暗-亮混合，球形示例完全显式，对称二聚体表现出相应的对称破缺交叉。
英文摘要
We establish a uniform pole-subtracted limiting absorption principle for a fixed cluster of \(N\) disjoint three-dimensional high-contrast elastic resonators when the contrast tends to infinity and \(ω=δ^{1/2}τ\) approaches the zero threshold. The exterior Dirichlet-to-Neumann map and a variational Grushin--Feshbach reduction give an exact decomposition of the cutoff resolvent into a uniformly bounded regular part and a finite-rank term governed by \[ \cM_δ^\pm(ω) = δK-ω^2I_m \mp\iiδωΓ_0 +\mathcal O(δ^2+δω^2). \] The cutoff-resolvent norm is uniformly equivalent to \(1+\|(\cM_δ^\pm(ω))^{-1}\|\); hence the finite-dimensional channel carries every loss of uniformity. An elastic optical identity factors the leading radiation matrix through one total-force map into \(\C^3\). Thus \(\rankΓ_0=3\) and \(\dim\KerΓ_0=6N-3\). Compression to a static eigenspace of dimension \(r\) leaves at most three leading radiative channels. Simple bright poles have width \(O(δ)\), whereas force-dark poles with a positive second radiation form have width \(O(δ^2)\); the corresponding real-axis peaks have orders \(δ^{-3/2}\) and \(δ^{-5/2}\). We also treat multiple static eigenvalues, compute the spherical coefficients, and derive a conditional two-parameter crossover for a symmetry-broken dimer.
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