Yury Shestopalov: Wave propagation and interaction phenomena: Introduction to Mathematical Theory

We give a short introduction to the spectral theory of open structures including some mathematical aspects of modeling the wave propagation in guides (cylindrical tubes) and interaction of oscillations and waves in electromagnetics and acoustics. The problems in question are considered in terms of the analysis of the spectral and various critical points (CPs) of multi-parameter operator-valued functions (OVFs), in particular, integral and infinite-matrix (summation) OVFs, when some of the operator parameters are varied. Interaction of oscillations in a cylindrical slotted resonator whose cross section is formed by two rectangular domains is taken as an example. We reduce the initial boundary eigenvalue problem for the Helmholtz equation or Maxwell equations to Fredholm integral equations with logarithmic singularity or to infinite systems with respect to the unknown Fourier coefficients of the solution. It turns out that some CPs of the OVF are associated with the points where one or several eigenvalues of partial domains merge. We demonstrate that interaction of oscillations occur in this case; namely, the electromagnetic field distributions become unstable with respect to small variations of certain parameters of the structure (geometric, permittivity etc.) in the vicinities of critical values.