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A real analyticity result for symmetric functions of the eigenvalues of a quasiperiodic spectral problem for the Dirichlet Laplacian

As is well known, by the Floquet--Bloch theory for periodic problems, one can transform a spectral Laplace--Dirichlet problem in the plane with a set of periodic perforations into a family of ``model problems'' depending on a parameter η∈[0,2π]2 for quasiperiodic functions in the unit cell with a single perforation. We prove real analyticity results for the eigenvalues of the model problems upon perturbation of the shape of the perforation of the unit~cell.