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Spectrum of a singularly perturbed periodic thin waveguide. (English) Zbl 1372.35195

Summary: We consider a family \(\Omega^\varepsilon_{\varepsilon > 0}\) of periodic domains in \(\mathbb{R}^2\) with waveguide geometry and analyse spectral properties of the Neumann Laplacian \(-\Delta_{{\Omega}^\varepsilon}\) on \(\Omega^\varepsilon\). The waveguide \(\Omega^\varepsilon\) is a union of thin straight strips of width \(\varepsilon\) and a family of small protuberances with the so-called “room-and-passage” geometry. The protuberances are attached periodically, with a period \(\varepsilon\), along the strip upper boundary. We prove a (kind of) resolvent convergence of \(-\Delta_{\Omega^\varepsilon}\) to a certain operator on the line as \(\varepsilon \rightarrow 0\). Also we demonstrate Hausdorff convergence of the spectrum. In particular, we conclude that if the sizes of “passages” are appropriately scaled the first spectral gap of \(-\Delta_{\Omega^\varepsilon}\) is determined exclusively by geometric properties of the protuberances. The proofs are carried out using methods of homogenization theory.

MSC:

35P15 Estimates of eigenvalues in context of PDEs
35P25 Scattering theory for PDEs
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