Some properties of functional-differential operators with involution $\nu(x)=1-x$ and their applications

Functional-differential operators with involution $\nu(x)=1-x$ are related to integral operators whose kernels suffer discontinuities on the lines $t=x$ and $t=1-x$, and to Dirac and Sturm-Liouville operators. They have found their application in the study of these operators, and in various applications. This paper reviews studies of 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.