Spectrum of the Laplacian on a domain perturbed by small resonators

TL;DR

This paper analyzes how small resonators attached to a domain affect the spectrum of the Laplacian, showing convergence to a union of the original spectrum and specific eigenvalues related to the resonators.

Contribution

It provides a rigorous analysis of the spectral convergence for Laplacians on domains perturbed by small resonators, including explicit eigenvalue limits and convergence rates.

Findings

Spectrum converges to the union of original spectrum and resonator eigenvalues.

Eigenvalues below the essential spectrum can be prescribed by resonator design.

Rate of convergence is estimated using Hausdorff-type metrics.

Abstract

It is widely known that the spectrum of the Dirichlet Laplacian is stable under small perturbations of a domain, while in the case of the Neumann or mixed boundary conditions the spectrum may abruptly change. In this work we discuss an example of such a domain perturbation. Let $\Omega$ be a (not {necessarily} bounded) domain in $\mathbb{R}^n$. We perturb it to $ \Omega_\varepsilon=\Omega\setminus \cup_{k=1}^m S_{k,\varepsilon},$ where $S_{k,\varepsilon}$ are closed surfaces with small suitably scaled holes (``windows'') through which the bounded domains enclosed by these surfaces (``resonators'') are connected to the outer domain. When $\varepsilon$ goes to zero, the resonators shrink to points. We prove that in the limit $\varepsilon\to 0$ the spectrum of the Laplacian on $\Omega_\varepsilon$ with the Neumann boundary conditions on $S_{k,\varepsilon}$ and the Dirichlet boundary conditions on the outer boundary converges to the union of the spectrum of the Dirichlet Laplacian on $\Omega$ and the numbers $\gamma_k$, $k=1,\dots,m$, being equal $1/4$ times the limit of the ratio between the capacity of the $k$th window and the volume of the $k$th resonator. We obtain an estimate on the rate of this convergence with respect to the Hausdorff-type metrics. Also, an application of this result is presented: we construct an unbounded waveguide-like domain with inserted resonators such that the eigenvalues of the Laplacian on this domain lying below the essential spectrum threshold do coincide with prescribed numbers.