Scattering Coefficients and Threshold Resonances in a Waveguide with Uniform Inflation of the Resonator

The spectral Dirichlet problem is considered in a waveguide made of a semi-infinite cylinder Π and the resonator Θ_R obtained by inflating R times a fixed star-shaped domain Θ. The behavior of the scattering coefficient s(R) is studied as the parameter R grows, namely, it is verified that this coefficient moves clockwise without stops along the unit circle in the complex plane. For s(R) = -1, the proper threshold resonance occurs; it is accompanied by the appearance of an almost standing wave and provokes various near-threshold anomalies, in particular, splitting eigenvalues off from the threshold. Under the geometrical symmetry, resonances of other type are shown to be generated by trapped waves at the threshold. The justification of asymptotics is made by applying the technique of weighted spaces with detached asymptotics and an analysis of the singularities of physical fields at the edge ∂Θ_R ∩ ∂Π.