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

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