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

¹ Inria, Ensta Paris, Institut Polytechnique de Paris, 828 Boulevard des Maréchaux, Palaiseau 91762, France
² Institute of Problems of Mechanical Engineering RAS, V.O., Bolshoi pr., 61, St. Petersburg 199178, Russia

^* Corresponding author: lucas.chesnel@inria.fr

Received: 18 January 2025
Accepted: 14 June 2025

Abstract

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.