We consider the leaky wave problem for an open inhomogeneous metal–dielectric circular-section waveguide. The problem is reduced to a boundary value problem for the longitudinal components of the electromagnetic field in Sobolev spaces. The inhomogeneity of the dielectric filling and the fact that the spectral parameter occurs in the matching conditions necessitate defining the solution of the problem in a special way. To define the solution, we use a variational statement of the problem. The variational problem is reduced to studying an operator function. The properties of the operator function needed for the analysis of its spectral characteristics are studied. Theorems on the discreteness of the spectrum and on the distribution of eigenvalues of the operator function on the complex plane are proved.

中文翻译：

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开放的非均匀金属介电圆截面波导漏波谱的离散性

我们考虑开放的非均匀金属介电圆形截面波导的漏波问题。该问题简化为 Sobolev 空间中电磁场纵向分量的边值问题。介电填充的不均匀性以及光谱参数出现在匹配条件中的事实需要以特殊方式定义问题的解决方案。为了定义解决方案，我们使用问题的变分陈述。变分问题简化为研究算子函数。研究了分析其光谱特性所需的算子函数的性质。证明了谱的离散性和算子函数在复平面上的特征值分布的定理。