Functional-differential operators with involution \(\nu(x)=1-x\), related to integral operators whose kernels can have points of discontinuity on the lines \(t=x\) and \(t=1-x\) and to Dirac and Sturm–Liouville operators, are used in the study of these operators and various applications. This paper provides a survey on the spectral properties of such operators with involution and their applications in problems on geometric graphs, in the study of Dirac systems, and in the justification of the Fourier method in mixed problems for partial differential equations.

中文翻译：

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对合$\nu(x)=1-x$的泛函微分算子的一些性质及其应用

具有对合\(\nu(x)=1-x\) 的函数微分运算符，与积分运算符有关，其内核可以在\(t=x\)和 \(t=1-x\ )线上具有不连续点)以及 Dirac 和 Sturm-Liouville 算子，用于研究这些算子和各种应用。本文综述了此类算子的谱特性及其在几何图问题、狄拉克系统研究以及偏微分方程混合问题中傅立叶方法的证明中的应用。