Creating and controlling band gaps in periodic media with small resonators
We investigate spectral properties of the Neumann Laplacian ${\mathcal A}_\varepsilon$ on a periodic unbounded domain ${\Omega}_\varepsilon$ depending on a small parameter $\varepsilon>0$. The domain ${\Omega}_\varepsilon$ is obtained by removing from ${\mathbb R}^n$ $m\in{\mathbb N}$ familie...
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| Date: | 2023 |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна Національної академії наук України
2023
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| Subjects: | |
| Online Access: | https://jmag.ilt.kharkiv.ua/index.php/jmag/article/view/1015 |
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| Journal Title: | Journal of Mathematical Physics, Analysis, Geometry |
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Journal of Mathematical Physics, Analysis, Geometry| Summary: | We investigate spectral properties of the Neumann Laplacian ${\mathcal A}_\varepsilon$ on a periodic unbounded domain ${\Omega}_\varepsilon$ depending on a small parameter $\varepsilon>0$. The domain ${\Omega}_\varepsilon$ is obtained by removing from ${\mathbb R}^n$ $m\in{\mathbb N}$ families of $\varepsilon$-periodically distributed small resonators. We prove that the spectrum of ${\mathcal A}_\varepsilon$ has at least $m$ gaps. The first $m$ gaps converge as $\varepsilon\to 0$ to some intervals whose location and lengths can be controlled by a suitable choice of the resonators; other gaps (if any) go to infinity. An application to the theory of photonic crystals is discussed.
Mathematical Subject Classification 2020: 35B27, 35P05, 47A75 |
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