Gaps in the spectrum of the Neumann Laplacian generated by a system of periodically distributed traps. (17th January 2014) - Khrabustovskyi, Andrii; Khruslov, Evgeni
<x xml:space="preserve">Abstract</x> The article deals with a convergence of the spectrum of the Neumann Laplacian in a periodic unbounded domain Ωϵ depending on a small parameter ϵ > 0. The domain has the form <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh2qt16rz2" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />Ωϵ=Rn∖Sϵ</alternatives>, where Sϵ is an <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh2qt16s1p" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />ϵZn</alternatives>‐periodic family of trap‐like screens. We prove that, for an arbitrarily large L, the spectrum has precisely one gap in [0, L] when ϵ is small enough; moreover, when ϵ → 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. Copyright © 2014 John Wiley & Sons, Ltd.