A waveguide in which the essential spectrum of the Laplace-Dirichlet problem consists of a countable set of points in the real positive semiaxis is constructed. The waveguide is built from a family of identical cells, which are connected by apertures in their common walls, while the sizes of the apertures decrease with distance from the "central" cell. It is shown that the lowest point of the essential spectrum is the limit of an infinite sequence of eigenvalues in the discrete spectrum. A hypothesis on the structure of the discrete spectrum inside gaps is stated, and some unsolved problems are mentioned. Bibliography: 8 titles. © 2013 Springer Science+Business Media New York.
Nazarov, S. A., & Taskinen, J. (2013). Structure of the Spectrum of a Periodic Family of Identical Cells Connected by Converging Apertures. Journal of Mathematical Sciences, 194(1), 72-82. https://doi.org/10.1007/s10958-013-1508-1