Tuesday, January 15, 2013

1301.2926 (Andrii Khrabustovskyi et al.)

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

Andrii Khrabustovskyi, Evgeni Khruslov
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.
View original: http://arxiv.org/abs/1301.2926

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