Opening up and control of spectral gaps of the Laplacian in periodic domains

The main result of this work is as follows: for arbitrary pairwise disjoint, finite intervals (alpha(j), alpha(j)) subset of [0,infinity), j = 1,..., m, and for arbitrary n >= 2, we construct a family of periodic non-compact domains {Omega(epsilon) subset of R-n} epsilon>0 such that the spectrum of the Neumann Laplacian in Omega(epsilon) has at least m gaps when e is small enough, moreover the first m gaps tend to the intervals (alpha(j), beta(j)) as epsilon -> 0. The constructed domain Omega(epsilon) is obtained by removing from R-n a system of periodically distributed "trap-like" surfaces. The parameter epsilon characterizes the period of the domain Omega(epsilon), also it is involved in a geometry of the removed surfaces. (C) 2014 AIP Publishing LLC.