Spectrum of the Laplacian on a Domain Perturbed by Small Resonators
Giuseppe Cardone, Andrii Khrabustovskyi · SIAM Journal on Mathematical Analysis · 2023
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 [Formula: see text] be a (not necessarily bounded) domain in [Formula: see text]. We perturb it to [Formula: see text] where [Formula: see text] 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 [Formula: see text] goes to zero, the resonators shrink to points. We prove that in the limit [Formula: see text] the spectrum of the Laplacian on [Formula: see text] with the Neumann boundary conditions on [Formula: see text] and the Dirichlet boundary conditions on the outer boundary converges to the union of the spectrum of the Dirichlet Laplacian on [Formula: see text] and the numbers [Formula: see text], [Formula: see text], being equal to [Formula: see text] times the limit of the ratio between the capacity of the [Formula: see text]th window and the volume of the [Formula: see text]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 the prescribed numbers.