On the breathing of spectral bands in periodic quantum waveguides with inflating resonators

Abstract

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.

Figures (12)

• Figure 1: Geometries of $\Omega$ (top left), $\omega^\varepsilon$ (top right) and $\Pi^\varepsilon$ (bottom).
• Figure 2: Dispersion curves (\ref{['Dispersion1']}) for $\theta=0$ (left), $\theta=\pi/8$ (center), $\theta=\pi/4$ (right).
• Figure 3: Sets $\Xi^\rho_m(\theta)$ (in magenta and blue) with respect to $\rho$. We use the magenta colour to stress the situations where $\Xi^\rho_m(\theta)$ take negative values. The vertical red dashed lines correspond to the $m^2\pi^2$, $m\in\mathbb{N}^{\ast}$. Here we take $T=2$ (arbitrarily) and $\sin(2\theta)=0.7$.
• Figure 4: Same quantities as in Figure \ref{['ImageAsymptoRho']} but this time with $\sin(2\theta)=1$.
• Figure 5: Geometries of $\Omega^H$ (left) and $\Omega^H_{1/2}$ (right).

Theorems & Definitions (17)

• Definition 3.1
• Definition 3.2
• Definition 3.3
• Proposition 3.4
• Theorem 3.5
• Remark 4.1
• Lemma 5.1
• proof
• Remark 5.2
• Remark 5.3