On the breathing of spectral bands in periodic quantum waveguides with inflating resonators
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914626017427456 |
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| author | Chesnel, Lucas Nazarov, Sergei A. |
| author_facet | Chesnel, Lucas Nazarov, Sergei A. |
| contents | We are interested in the lower part of the spectrum of the Dirichlet Laplacian $A^\varepsilon$ in a thin waveguide $Π^\varepsilon$ obtained by repeating periodically a pattern, itself constructed by scaling an inner field geometry $Ω$ by a small factor $\varepsilon>0$. The Floquet-Bloch theory ensures that the spectrum of $A^\varepsilon$ has a band-gap structure. Due to the Dirichlet boundary conditions, these bands all move to $+\infty$ as $O(\varepsilon^{-2})$ when $\varepsilon\to0^+$. Concerning their widths, applying techniques of dimension reduction, we show that the results depend on the dimension of the so-called space of almost standing waves in $Ω$ that we denote by $\mathrm{X}_\dagger$. Generically, i.e. for most $Ω$, there holds $\mathrm{X}_\dagger=\{0\}$ and the lower part of the spectrum of $A^\varepsilon$ is very sparse, made of bands of length at most $O(\varepsilon)$ as $\varepsilon\to0^+$. For certain $Ω$ however, we have $\mathrm{dim}\,\mathrm{X}_\dagger=1$ and then there are bands of length $O(1)$ which allow for wave propagation in $Π^\varepsilon$. The main originality of this work lies in the study of the behaviour of the spectral bands when perturbing $Ω$ around a particular $Ω_\star$ where $\mathrm{dim}\,\mathrm{X}_\dagger=1$. We show a breathing phenomenon for the spectrum of $A^\varepsilon$: when inflating $Ω$ around $Ω_\star$, the spectral bands rapidly expand before shrinking. In the process, a band dives below the normalized threshold $π^2/\varepsilon^2$, stops breathing and becomes extremely short as $Ω$ continues to inflate. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_00439 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the breathing of spectral bands in periodic quantum waveguides with inflating resonators Chesnel, Lucas Nazarov, Sergei A. Analysis of PDEs Mathematical Physics We are interested in the lower part of the spectrum of the Dirichlet Laplacian $A^\varepsilon$ in a thin waveguide $Π^\varepsilon$ obtained by repeating periodically a pattern, itself constructed by scaling an inner field geometry $Ω$ by a small factor $\varepsilon>0$. The Floquet-Bloch theory ensures that the spectrum of $A^\varepsilon$ has a band-gap structure. Due to the Dirichlet boundary conditions, these bands all move to $+\infty$ as $O(\varepsilon^{-2})$ when $\varepsilon\to0^+$. Concerning their widths, applying techniques of dimension reduction, we show that the results depend on the dimension of the so-called space of almost standing waves in $Ω$ that we denote by $\mathrm{X}_\dagger$. Generically, i.e. for most $Ω$, there holds $\mathrm{X}_\dagger=\{0\}$ and the lower part of the spectrum of $A^\varepsilon$ is very sparse, made of bands of length at most $O(\varepsilon)$ as $\varepsilon\to0^+$. For certain $Ω$ however, we have $\mathrm{dim}\,\mathrm{X}_\dagger=1$ and then there are bands of length $O(1)$ which allow for wave propagation in $Π^\varepsilon$. The main originality of this work lies in the study of the behaviour of the spectral bands when perturbing $Ω$ around a particular $Ω_\star$ where $\mathrm{dim}\,\mathrm{X}_\dagger=1$. We show a breathing phenomenon for the spectrum of $A^\varepsilon$: when inflating $Ω$ around $Ω_\star$, the spectral bands rapidly expand before shrinking. In the process, a band dives below the normalized threshold $π^2/\varepsilon^2$, stops breathing and becomes extremely short as $Ω$ continues to inflate. |
| title | On the breathing of spectral bands in periodic quantum waveguides with inflating resonators |
| topic | Analysis of PDEs Mathematical Physics |
| url | https://arxiv.org/abs/2401.00439 |