## 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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### References:

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