On the sharpneess of a certain spectral stability estimate for the Dirichlet Laplacian

We consider a spectral stability estimate by Burenkov and Lamberti concerning the variation of the eigenvalues of second order uniformly elliptic operators on variable open sets in the $N$-dimensional euclidean space, and we prove that it is sharp for any dimension $N$. This is done by studying the eigenvalue problem for the Dirichlet Laplacian on special open sets inscribed in suitable spherical cones.