Sergei A. Nazarov: Asymptotic analysis of spectral gaps in periodic waveguides

The spectral gap opening will be demonstrated in waveguides perforated with a periodic family of small holes and in a plane with double periodic family of big circular holes. The asymptotic origin of opening gaps is of different nature for these two problems, however both the cases an asymptotic analysis gives explicit formulas for the geometrical characteristics of spectral bands and gaps. 

The demonstration is given for the Neumann Laplacian (e.g., acoustic waveguides) but generalizations for other boundary conditions and systems, for example elasticity, will be discussed as well. Open question will be indicated, too. The solution of the main question in the band-gap engineering will be given, namely any given point of the spectrum in a cylindrical waveguide can be put inside a spectral gap under a smal perturbation of the waveguide.