Structure of the spectrum of the periodic family of identical cells connected through apertures of reducing sizes

A waveguide is constructed such that the Dirichlet problem for the Laplace operator gets the essential spectrum implying a countable set of points in the real positive semi-axis. The waveguide is obtained by joining identical cells through apertures in their common walls. Size of the apertures decreases at distance from the “central” cell. It is shown that the first point of the essential spectrum is a limit of an infinite sequence of eigenvalues of the problem from its discrete spectrum. A hypothesis on the structure of the discrete spectrum inside spectral gaps is formulated and other open questions are mentioned.