Spectrum of the Laplacian on a Domain Perturbed by Small Resonators

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 Ω be a (not necessarily bounded) domain in ℝn. We perturb it to (Equation presented), where Sk, ε 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 ε goes to zero, the resonators shrink to points. We prove that in the limit ε → 0 the spectrum of the Laplacian on Ωε with the Neumann boundary conditions on Sk, ε and the Dirichlet boundary conditions on the outer boundary converges to the union of the spectrum of the Dirichlet Laplacian on Ω and the numbers γk, k = 1, ..., m, being equal to 1/4 times the limit of the ratio between the capacity of the kth window and the volume of the kth 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.