Barrier composed of perforated resonators and boundary conditions

We consider the Laplace operator with the Neumann boundary condition in a two-dimensional domain divided by a barrier composed of many small Helmholtz resonators coupled with the both parts of the domain through small windows of diameter $2a$. The main terms of the asymptotic expansions in a of the eigenvalues and eigenfunctions are considered in the case in which the number of the Helmholtz resonators tends to innity. It is shown that such a homogenization procedure leads to some energy-dependent boundary condition in the limit. We use the method of matching the asymptotic expansions of boundary value problem solutions.