Scattering coefficients and threshold resonances in a waveguide with inflating resonator

The spectral Dirichlet problem is considered in a waveguide formed from a semi-infinite cylinder $\Pi$ and the resonator $\Theta_R$ obtained by inflating $R$ times a fixed star-shaped domain $\Theta$. The behaviour 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 which is accompanied by the appearance of an almost standing wave and provokes for various near threshold anomalies, in particular, splitting eigenvalues off from the threshold. It is shown that under the geometrical symmetry resonances of other type are 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 $\partial\Theta_R\cap \partial \Pi$.