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

Abstract

In this note we investigate spectral properties of a periodic waveguide Ω ε (ε is a small parameter) obtained from a straight strip by attaching an array of ε-periodically distributed identical protuberances having "room-and-passage" geometry. In the current work we consider the operator A ε = −ρ ε ∆Ωε , where ∆Ωε is the Neumann Laplacian in Ω ε , the weight ρ ε is equal to 1 everywhere except the union of the "rooms". We will prove that the spectrum of A ε has at least one gap as ε is small enough provided certain conditions on the weight ρ ε and the sizes of attached protuberances hold.

Spectrum of a singularly perturbed periodic thin waveguide

“…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. Moreover, in contrast to [12], both edges of this gap depends on geometric properties of the waveguide in a very simple fashion.…”

“…The results of [11] were extended in [12] 1 to Ω, which is an unbounded straight strip of the fixed width L > 0. In this case Γ is its upper (or lower) boundary.…”

“…In this case Γ is its upper (or lower) boundary. It turns out that the form of the limit problem remains the same as in the case of a bounded domain, but the structure of its spectrum is essentially different: it is either the whole positive semi-axis or the set [0, ∞) \ (q, q) (this case occur 1 In fact, in [12] we deal with the most interesting case q > 0, lim ). The number q ∈ (q, ∞) is a solutions to some transcendental equation involving q and L. In the last instance we are able to make the following useful conclusion: the spectrum of H ε has a gap provided ε is small enough, the edges of this gap converge to q and q.…”

“…In the current paper we study the asymptotic behaviour of the "pure" Neumann Laplacian (i.e., ε = 1), but now, in contrast to [12], the "basic" strip Ω = Π ε also depends on ε.…”