发表机构
ETH Zürich; Hong Kong Institute for Advanced Study, City University of Hong Kong; Qiuzhen College, Tsinghua University; School of Science, Institute for Theoretical Sciences, Westlake University; Yau Mathematical Sciences Center, Tsinghua University; Beijing Institute of Mathematical Sciences and Applications（苏黎世联邦理工学院; 香港城市大学高等研究院; 清华大学求真书院; 西湖大学理学院理论科学研究所; 清华大学丘成桐数学科学中心; 北京国际数学研究中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究针对有限气泡晶体，证明了第一布洛赫带上边缘的定量声软散射极限，通过均匀化方法获得O(ε)近似，并揭示内部场收敛阶数及一维透射窗口的非唯一性。
AI 中文摘要
我们建立了有限气泡晶体在第一布洛赫带上边缘附近的定量声软散射极限，该晶体对应于无限晶体。夹杂物在有界Lipschitz域中形成周期为$\varepsilon$的稠密周期阵列，其密度对比度为$\delta=\varepsilon^2$，波速固定为正。在坐标反射对称性下，我们证明了归一化电容符号在布里渊环面上具有唯一的非退化最大值。其Hessian矩阵定义了一个正Dirichlet椭圆算子，控制极限光谱失谐。一个均值约束变分公式使我们能够将实际有限阵列电容矩阵与截断的无限晶格算子进行比较，并证明它们的差异在样本边界附近呈指数局部化。我们获得了$O(\varepsilon)$范数预解近似，并将其与完整声学散射问题联系起来，保留了频率缩放所需的二阶Bloch修正。对于有效Dirichlet谱之外的重新缩放失谐，外部$L^2$和远场差异为$O(\varepsilon)$，而内部$L^2$场为$O(\varepsilon^{1/2})$。数值实验说明了远场收敛和有效光谱模式。显式一维计算描述了更精细的Fabry–Pérot透射窗口，并表明严格位于第一带内的固定频率不一定具有唯一的散射极限。
英文摘要
We establish a quantitative sound-soft scattering limit for a finite bubbly crystal near the upper edge of the first Bloch band of the corresponding infinite crystal. The inclusions form a dense periodic array of period $\varepsilon$ in a bounded Lipschitz domain, and their density contrast is $δ=\varepsilon^2$, with fixed positive wave speeds. Under coordinate-reflection symmetry, we prove that the normalized capacitance symbol has a unique nondegenerate maximum on the Brillouin torus. Its Hessian defines a positive Dirichlet elliptic operator governing the limiting spectral detunings. A mean-constrained variational formulation allows us to compare the actual finite-array capacitance matrix with the truncated infinite-lattice operator and to prove exponential localization of their difference near the sample boundary. We obtain an $O(\varepsilon)$ norm-resolvent approximation and connect it to the full acoustic scattering problem, retaining the second-order Bloch correction required by the frequency scaling. For rescaled detunings outside the effective Dirichlet spectrum, the exterior $L^2$ and far-field discrepancies are $O(\varepsilon)$, while the interior $L^2$ field is $O(\varepsilon^{1/2})$. Numerical experiments illustrate the far-field convergence and the effective spectral modes. Explicit one-dimensional calculations describe finer Fabry--Pérot transmission windows and show that a fixed frequency strictly inside the first band need not have a unique scattering limit.
Comments75 pages, 7 figures
