Gaps in the spectrum of the Neumann Laplacian generated by a system of periodically distributed trap

TL;DR

This paper studies how the spectrum of the Neumann Laplacian in a periodic domain with trap-like screens develops gaps as the scale parameter shrinks, with implications for designing photonic crystals.

Contribution

It demonstrates the existence and controllability of spectral gaps in the Neumann Laplacian spectrum for small periodic traps, advancing understanding of spectral properties in complex domains.

Findings

Spectrum has exactly one gap in [0,L] for small ε

The gap converges to a controllable interval as ε→0

Application to 2D-photonic crystal theory

Abstract

The article deals with a convergence of the spectrum of the Neumann Laplacian in a periodic unbounded domain $\Omega^\varepsilon$ depending on a small parameter $\varepsilon>0$. The domain has the form $\Omega^\varepsilon=\mathbb{R}^n\setminus S^\varepsilon$, where $S^\varepsilon$ is an $\varepsilon\mathbb{Z}^n$-periodic family of trap-like screens. We prove that for an arbitrarily large $L$ the spectrum has just one gap in $[0,L]$ when $\varepsilon$ small enough, moreover when $\varepsilon\to 0$ this gap converges to some interval whose edges can be controlled by a suitable choice of geometry of the screens. An application to the theory of 2D-photonic crystals is discussed.