Abstract

The problem on normal TE-waves in an inhomogeneous partially shielded dielectric layer is reduced to eigenvalue problem for the component of the electromagnetic field in Sobolev space. We formulate the definition of solution using variational relation. The variational problem is reduced to the study of an operator pencil. We investigate properties of the operators of the pencil for the analysis of its spectral properties. We prove theorem of discrete spectrum and theorem of localization of eigenvalues of the operator pencil on complex plane. The main result of the paper is the proof of double completeness of eigen- and associated vectors of the pencil in Sobolev space, which leads to existence of infinitely many waves in the waveguiding structure.

中文翻译：

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关于非均匀部分屏蔽介电层中法向波的完备性

摘要

非均匀部分屏蔽介电层中的正常 TE 波问题被简化为 Sobolev 空间中电磁场分量的特征值问题。我们使用变分关系来制定解决方案的定义。变分问题被简化为操作员铅笔的研究。我们研究铅笔算子的性质以分析其光谱性质。我们证明了离散谱定理和复平面算子铅笔特征值的局部化定理。论文的主要结果是证明了 Sobolev 空间中铅笔的特征向量和相关向量的双重完备性，这导致波导结构中存在无限多的波。