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

Abstract

The main result of this work is as follows: for arbitrary pairwise disjoint finite intervals $(α_j,β_j)\subset[0,\infty)$, $j=1,\dots,m$ and for arbitrary $n\geq 2$ we construct the family of periodic non-compact domains $\{Ω^\varepsilon\subset\mathbb{R}^n\}_{\varepsilon>0}$ such that the spectrum of the Neumann Laplacian in $Ω^\varepsilon$ has at least $m$ gaps when $\varepsilon$ is small enough, moreover the first $m$ gaps tend to the intervals $(α_j,β_j)$ as $\varepsilon\to 0$. The constructed domain $Ω^\varepsilon$ 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 $Ω^\varepsilon$, also it is involved in a geometry of the removed surfaces.