On the properties of phononic eigenvalue problems
On the properties of phononic eigenvalue problems
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DOI：
10.1016/j.jmps.2019.07.005
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发表时间：
2019-02
影响因子：
5.3
通讯作者：
A. Mokhtari;A. Srivastava
中科院分区：
文献类型：
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作者：
A. Mokhtari;A. Srivastava
本文考虑了各种声子本征值问题的算子性质。我们的目的是回答一些基本问题的声子算符的本征值和本征向量。这些问题包括潜在的真实的和复杂的性质的特征值，是否特征向量形成一个完整的基础，什么是正确的正交关系，以及如何创建一个完整的基础时，可能不存在在一开始。在这样做的时候，我们提出了一个统一的理解的声子本征值和本征向量，将出现从任何数值方法来计算这些数量的属性。我们表明，声子问题可以被铸造成线性本征值的形式，从这些数量的频率，波数，和所需的组件的波矢量可以直接确定，而不诉诸搜索或二次本征值问题，这些数量的相关属性可以确定先验通过相关运营商的分析。我们进一步展示了如何平面波展开（PWE）方法可以扩展到解决这些特征值的形式，从而扩展的PWE方法的适用性超出那些已经被认为是到现在为止的情况。理论讨论的补充支持数值计算。这里提出的技术和结果直接适用于波在其他周期性系统，如光子学的传播。
In this paper, we consider the operator properties of various phononic eigenvalue problems. We aim to answer some fundamental questions about the eigenvalues and eigenvectors of phononic operators. These include questions about the potential real and complex nature of the eigenvalues, whether the eigenvectors form a complete basis, what are the right orthogonality relationships, and how to create a complete basis when none may exist at the outset. In doing so we present a unified understanding of the properties of the phononic eigenvalues and eigenvectors which would emerge from any numerical method employed to compute such quantities. We show that the phononic problem can be cast into linear eigenvalue forms from which such quantities as frequencies, wavenumbers, and desired components of wavevectors can be directly ascertained without resorting to searches or quadratic eigenvalue problems and that the relevant properties of such quantities can be determined apriori through the analysis of the associated operators. We further show how the Plane Wave Expansion (PWE) method may be extended to solve each of these eigenvalue forms, thus extending the applicability of the PWE method to cases beyond those which have been considered till now. The theoretical discussions are supplemented with supporting numerical calculations. The techniques and results presented here directly apply to wave propagation in other periodic systems such as photonics.