Spectral gaps for the Dirichlet‐Laplacian in a 3‐D waveguide periodically perturbed by a family of concentrated masses

Abstract

We consider a spectral problem for the Laplace operator in a periodic waveguide Π⊂R3 perturbed by a family of “heavy concentrated masses”; namely, Π contains small regions false{ωjεfalse}j∈double-struckZ of high density, which are periodically distributed along the z axis. Each domain ωjε⊂Π has a diameter O(ε) and the density takes the value ε−m in ωjε and 1 outside; m and ε are positive parameters, m>2, ε≪1. Considering a Dirichlet boundary condition, we study the band‐gap structure of the essential spectrum of the corresponding operator as ε→0. We provide information on the width of the first bands and find asymptotic formulas for the localization of the possible gaps.

Band-gap structure of the spectrum of the water-wave problem in a shallow canal with a periodic family of deep pools

“…Finally, in Sect. 4 we consider the limit problem in more detail under some additional assumptions. We introduce a new geometric parameter ε > 0 and observe that, for small enough ε, the limit problem has a spectral gap.…”

“…Of course, neither the present paper nor [7] are the first studies of spectral gaps by means of asymptotic analysis. Let us mention [3,4,6,20,23], where the detection of open gaps is based on a periodic perforation of strips and cylinders or singular perturbations of a similar type, an approach which will also be used in Sect. 4.…”

Spectral gaps in a double-periodic perforated Neumann waveguide

“…There are numerous publications in which open spectral gaps are detected due to high-contrast of coefficients in differential operators or shape irregularities of the periodicity cells, see [15,16,45,29,5,4,3,6] and [26,34,39,35,7] among others. Such singular perturbations often provide disintegration of the periodicity cells in the limit and, as a result, the appearance of sufficiently wide gaps in the low-and/or middle-frequency ranges of the spectrum.…”

The stiff Neumann problem: asymptotic specialty and "kissing" domains

“…[2,8,9,10,30]). In the context of second order differential operator with double periodic coefficients, we also mention [5,6,15,16,34], where the authors investigate how to give rise to spectral gaps in the essential spectrum. In the last part of the paper, we handle the same stiff problem (1)-( 4) but with a geometry of the domain Ω, which differs from that drawn in Fig 1 . A irregular point appears on the boundary ∂Ω, consisting of the point O of tangency of the two "kissing" disks Ω0 and Ω1 in R 2 (see Fig.…”