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