Spectral gaps in a thin-walled infinite rectangular Dirichlet box with a periodic family of cross walls

Abstract

The Dirichlet spectral problem for the Laplace operator is considered in an infinite thin-walled rectangular box with a periodic family of cross walls whose thickness is proportional to that of the walls. Using asymptotic analysis it is shown that spectral gaps open up in the case of ‘thin’ or ‘sufficiently thick’ cross-walls whose relative thickness is bounded above or below by certain characteristics of model Dirichlet problems in $\mathsf L$- and $\mathsf T$-shaped domains in the plane and in a union of two pairwise orthogonal halves of space layers and a quarter of a space layer. A number of open questions are stated; in particular, because of the lack of information on threshold resonances in the three-dimensional model problem, the structure of the spectrum for cross walls of any intermediate thickness remains unknown. Bibliography: 35 titles.