Gaps in the Spectrum of Thin Waveguides with Periodically Locally Deformed Walls

The structures of quantum and acoustic waveguides obtained by joining a periodic family of small knots to a thin cylinder are examined. Asymptotic expansions of eigenvalues of a model problem in the periodicity cell are obtained, which are used to derive asymptotic formulas for the disposition and sizes of the gaps in the spectra of the corresponding Dirichlet and Neumann problems for the Laplace operator. Geometric and integral characteristics of the waveguides are found that ensure the opening of several spectral gaps.