The convergence of spectra via two-scale convergence for double-porosity models is well known. A crucial assumption in these works is that the stiff component of the body forms a connected set. We show that under a relaxation of this assumption the (periodic) two-scale limit of the operator is insufficient to capture the full asymptotic spectral properties of high-contrast periodic media. Asymptotically, waves of all periods (or quasi-momenta) are shown to persist and an appropriate extension of the notion of two-scale convergence is introduced. As a result, homogenised limit equations with none trivial quasi-momentum dependence are found as resolvent limits of the original operator family. This results in asymptotic spectral behaviour with a rich dependence on quasimomenta.

中文翻译：

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高对比度介质中的准周期两尺度均质化和有效空间分散。

通过双孔隙率模型的两尺度收敛的光谱收敛是众所周知的。这些工作中的一个关键假设是，身体的僵硬部分形成一个连接的集合。我们表明，在这种假设的放松下，算子的（周期性）两尺度极限不足以捕获高对比度周期性介质的全部渐近光谱特性。渐近地，所有周期（或准动量）的波都显示为持续存在，并引入了两尺度收敛的概念的适当扩展。结果，发现没有平凡准动量依赖性的均质极限方程作为原始算子族的分解极限。这导致了渐进光谱行为，对拟定门的依赖性很大。