Spectrum of a singularly perturbed periodic thin waveguide

2016/08/01 by Cardone, Giuseppe, Khrabustovskyi, Andrii
#35B27 #35PXX #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1608.00440

Abstract

We consider a family \Ω^ε\_ε>0 of periodic domains in ℝ² with waveguide geometry and analyse spectral properties of the Neumann Laplacian -Δ_Ω^ε on Ω^ε. The waveguide Ω^ε is a union of a thin straight strip of the width ε and a family of small protuberances with the so-called "room-and-passage" geometry. The protuberances are attached periodically, with a period ε, along the strip upper boundary. For ε→ 0 we prove a (kind of) resolvent convergence of -Δ_Ω^ε to a certain ordinary differential operator. Also we demonstrate Hausdorff convergence of the spectrum. In particular, we conclude that if the sizes of "passages" are appropriately scaled the first spectral gap of -Δ_Ω^ε is determined exclusively by geometric properties of the protuberances. The proofs are carried out using methods of homogenization theory.

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