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We analyze the behavior of the eigenvalues and eigenfunctions of the Laplaceoperator with homogeneous Neumann boundary conditions when the domain isperturbed. We focus on exterior perturbations of the domain, that is, the limitdomain is contained in each of the perturbed domains. Moreover, the family ofperturbed domains is made up of bounded domains although not necessarilyuniformly bounded. We show that if we know that the eigenvalues andeigenfunctions of the perturbed domains converge to the ones of the limitdomain, then necessarily we have that the measure of the perturbation tends tozero, while it is not necessarily true for the Hausdorff distance of theperturbation. As a matter of fact we will construct an example of aperturbation where the spectra behave continuously but the distance tends toinfinity.

Geometric versus spectral convergence for the Neumann Laplacian under exterior perturbations of the domain