Asymptotics of bound states and bands for laterally coupled three-dimensional waveguides

A system of two three-dimensional waveguides coupled laterally through a small window is considered. The asymptotics (in the width of window) of an eigenvalue close to the threshold is obtained. The waveguides coupled through periodic systems of windows are studied. The asymptotics of the band is obtained. The technique is the matching of asymptotic expansions of solutions.