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

TL;DR

This paper demonstrates how to design periodic domains with controllable spectral gaps for the Neumann Laplacian, by removing trap-like surfaces, allowing precise tuning of spectral properties as the domain's period shrinks.

Contribution

It constructs a family of periodic domains with spectral gaps that can be made to approximate arbitrary intervals, advancing control over spectral properties of Laplacians in complex geometries.

Findings

Spectral gaps can be engineered to approximate arbitrary intervals.

The domain construction involves periodically distributed trap-like surfaces.

The spectral gaps become more pronounced as the period parameter decreases.

Abstract

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.