Creating and controlling band gaps in periodic media with small resonators

2022/12/15 by Khrabustovskyi, Andrii, Khruslov, Evgen
#35B27 #35P05 #47A75 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2212.08100

Abstract

We investigate spectral properties of the Neumann Laplacian \mathscrA_ε on a periodic unbounded domain Ω_ε depending on a small parameter ε>0. The domain Ω_ε is obtained by removing from ℝⁿ m∈ℕ families of ε-periodically distributed small resonators. We prove that the spectrum of \mathscrA_ε has at least m gaps. The first m gaps converge as ε→ 0 to some intervals whose location and lengths can be controlled by a suitable choice the resonators; other gaps (if any) go to infinity. An application to the theory of photonic crystals is discussed.

Related

• Example of periodic Neumann waveguide with gap in spectrum
• Spectrum of the Laplacian on a domain perturbed by small resonators
• Spectrum of a singularly perturbed periodic thin waveguide
• Creating band gaps in periodic media
• A note on the spectrum of the Neumann Laplacian in periodic waveguides