Asymptotics of bound states and bands for waveguides coupled through small windows

A system of two waveguides coupled laterally through small windows is considered. The asymptotics (in the width of windows) of ground state close to the threshold is obtained for the case of finite number of apertures. The cases of periodic system of coupling windows is studied. The asymptotics of the band edges is obtained. The technique is matching of asymptotic expansions of the solutions.