Radiating edges and scattering by domains with infinitely narrow slots

Reference is made to earlier studies in which the operator expansion theory was used to develop a mathematical model of a hollow resonator providing a relatively simple way to calculate some spectral characteristics of the dissipative differential operator of the corresponding scattering problem. In the present study, it is shown that there exists a class of real diffraction problems for regions with small but finite openings of 2d diameter whose solutions tend to solutions based on the proposed model as d tends to zero. The applicability of the model to real problems is also discussed, from a different standpoint, with reference to a three-dimensional resonator with a small opening.