Localization of Waves and Their Stopping Zones in a Thin-Walled Hollow Quantum Waveguide with Periodic Cross-Walls

The spectral Dirichlet problem in a thin-walled cylinder with a periodic family of cross-walls that are perpendicular to the cylinder generatrix is considered. An asymptotic analysis of the problem establishes the existence of open gaps in the waveguide spectrum and yields the positions and lengths of the spectral bands. For cross-walls of different relative thicknesses, the eigenfunctions of a model problem on the periodicity cell with a Floquet parameter exhibit different types of localization. Namely, traveling waves concentrate at the cross-walls (large thickness) or near their edges (small thickness). The results are obtained by applying various procedures for dimension reduction and analysis of a boundary layer developing near the joints between the cross-walls and the cylinder. The boundary layer is described by the Dirichlet problem in a planar T-shaped junction of a unit strip and a perpendicular half-strip of different thickness. The localization manner is determined by whether or not the last problem has a discrete spectrum.