Resonator systems

The paper deals with a system of embedded resonators and a chain of two resonators. We prove that the Green functions of the corresponding Neumann boundary-value problems have poles with small imaginary parts. We find complete asymptotics for these poles and the corresponding eigenfunctions by the method of matched asymptotic expansions. We consider the cases when the limit value of the pole is an eigenfrequency either of a single limit volume or of two such volumes simultaneously. We show that the orders of smallness of the imaginary parts of the poles for systems are quite different from those for the classical Helmholtz resonator. We apply the asymptotics obtained to the scattering problem.