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 $A^{\epsilon }$ in a thin waveguide ${\Pi }^{\epsilon }$ obtained by repeating periodically a pattern, itself constructed by scaling an inner field geometry $\Omega$ by a small factor $\epsilon >0$. The Floquet-Bloch theory ensures that the spectrum of $A^{\epsilon }$ has a band-gap structure. Due to the Dirichlet boundary conditions, these bands all move to $+\infty$ as $O({\epsilon }^{-2})$ when $\epsilon \to 0^{+}$. 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 $\Omega$ that we denote by ${\mathrm{X}}_{†}$. Generically, i.e. for most $\Omega$, there holds ${\mathrm{X}}_{†}=\{0\}$ and the lower part of the spectrum of $A^{\epsilon }$ is very sparse, made of bands of length at most $O(\epsilon )$ as $\epsilon \to 0^{+}$. For certain $\Omega$ however, we have $\dim {\mathrm{X}}_{†}=1$ and then there are bands of length $O(1)$ which allow for wave propagation in ${\Pi }^{\epsilon }$. The main originality of this work lies in the study of the behaviour of the spectral bands when perturbing $\Omega$ around a particular ${\Omega }_{\star }$ where $\dim {\mathrm{X}}_{†}=1$. We show a breathing phenomenon for the spectrum of $A^{\epsilon }$: when inflating $\Omega$ around ${\Omega }_{\star }$, the spectral bands rapidly expand before shrinking. In the process, a band dives below the normalized threshold ${\pi }^2/{\epsilon }^2$, stops breathing and becomes extremely short as $\Omega$ continues to inflate.