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 $(\alpha_j,\beta_j)\subset[0,\infty)$, $j=1,\dots,m$ and for arbitrary $n\geq 2$ we construct the family of periodic non-compact domains $\{\Omega^\varepsilon\subset\mathbb{R}^n\}_{\varepsilon>0}$ such that the spectrum of the Neumann Laplacian in $\Omega^\varepsilon$ has at least $m$ gaps when $\varepsilon$ is small enough, moreover the first $m$ gaps tend to the intervals $(\alpha_j,\beta_j)$ as $\varepsilon\to 0$. The constructed domain $\Omega^\varepsilon$ is obtained by removing from $\mathbb{R}^n$ a system of periodically distributed "trap-like" surfaces. The parameter $\varepsilon$ characterizes the period of the domain $\Omega^\varepsilon$, also it is involved in a geometry of the removed surfaces.