非线性本征值问题及其在光子晶体中的应用
Non-linear eigenvalue problems and applications to photonic crystals
摘要：
我们对一类常见的有理算子Nevanlinna函数建立了新的分析和数值结果，这些函数出现在例如光子晶体建模中。这些介电纳米结构材料控制光的流动的能力取决于其本征值的具体特征。我们的结果提供了一个完整的谱分析，包括变分原理和双边估计的所有特征值以及数值实现。他们甚至适用于多极洛伦兹模型的介电常数函数和特征值之间的极点，经典的最小-最大变分原理完全失败。特别是，我们表明，我们的抽象的双边特征值估计是最佳的，我们推导出明确的界限上的带隙以上的洛伦兹极点。本文用高阶有限元方法计算了几种具体的洛仑兹模型（如极化激元材料和多极模型）的特征值的双边估计。
英文摘要：
We establish new analytic and numerical results on a general class of rational operator Nevanlinna functions that arise e.g. in modelling photonic crystals. The capability of these dielectric nano-structured materials to control the flow of light depends on specific features of their eigenvalues. Our results provide a complete spectral analysis including variational principles and two-sided estimates for all eigenvalues along with numerical implementations. They even apply to multi-pole Lorentz models of permittivity functions and to the eigenvalues between the poles where classical min-max variational principles fail completely. In particular, we show that our abstract two-sided eigenvalue estimates are optimal and we derive explicit bounds on the band gap above a Lorentz pole. A high order finite element method is used to compute the two-sided estimates of a selection of eigenvalues for several concrete Lorentz models, e.g. polaritonic materials and multi-pole models.
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