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 π²/ε², stops breathing and becomes extremely short as Ω continues to inflate. These results are illustrated by numerical experiments.