Gap Opening Around a Given Point of the Spectrum of a Cylindrical Waveguide by Means of a Gentle Periodic Perturbation of Walls

Назаров, С. А.

doi:10.1007/s10958-015-2312-x

Cited by 1 publication

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“…In this case the spectrum typically has small bands separated by relatively large gaps. The opposite picture (i.e., large bands versus small gaps) occurs under "small" perturbations of straight waveguides (see [4,14,25,27,28]) -either by a periodic nucleation of small holes or by a gentle periodic bending of the boundary. The waveguide Ω ε constructed in the current paper falls into an intermediate case -the length of the first band is comparable with the length of the first gap.…”

Section: Introductionmentioning

confidence: 99%

Spectrum of a singularly perturbed periodic thin waveguide

Cardone

Khrabustovskyi

2017

Journal of Mathematical Analysis and Applications

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We consider a family {Ω ε } ε>0 of periodic domains in R 2 with waveguide geometry and analyse spectral properties of the Neumann Laplacian −∆ 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 −∆ 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 −∆ 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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