BooksecrCollected Volumepp. 177–187

Example of a periodic Neumann waveguide with a gap in its spectrum

  • Giuseppe Cardone

    Università del Sannio, Benevento, Italy
  • Andrii Khrabustovskyi

    Karlsruher Institut für Technologie, Germany
Example of a periodic Neumann waveguide with a gap in its spectrum cover
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Abstract

In this note we investigate spectral properties of a periodic waveguide Ωε\Omega^\varepsilon (ε\varepsilon is a small parameter) obtained from a straight strip by attaching an array of ε\varepsilon-periodically distributed identical protuberances having "room-and-passage" geometry. In the current work we consider the operator Aε=ρεΔΩε\mathcal{A}^\varepsilon =-\rho^\varepsilon\Delta_{\Omega^\varepsilon}, where ΔΩε\Delta_{\Omega^\varepsilon} is the Neumann Laplacian in Ωε\Omega^\varepsilon, the weight ρε\rho^\varepsilon is equal to 11 everywhere except the union of the „rooms". We will prove that the spectrum of Aε\mathcal{A}^\varepsilon has at least one gap as ε\varepsilon is small enough provided certain conditions on the weight ρε\rho^\varepsilon and the sizes of attached protuberances hold.