On Scattering by a Cylindrical Trap in Critical Case

We consider a two-dimensional analogue of Helmholtz resonator with walls of finite thickness in the critical case when there exists an eigenfrequency equalling to the limit of poles generated by both the bounded component of the resonator and by the narrow connecting channel. Under assumption that the limit eigenfrequency is simple one of the bounded component, asymptotics of two poles converging to this eigenfrequency are constructed by using the method of matching asymptotic expansions. The explicit formulas for the leading terms of asymptotics for poles and for the solution of the scattering problem are obtained.