The effect of perturbations on the multiplicity of eigenvalues for the fractional Laplacian on bounded domain

We consider the Dirichlet eigenvalues of the fractional Laplacian related to a smooth bounded domain. We will prove that there exists an arbitrarily small perturbation of the original domain for which all Dirichlet eigenvalues of the fractional Laplacian are simple. Also, the same result of simplicity of eigenvalues holds for a generic perturbation of the coefficients of the eigenvalue equation. Finally we study the set of perturbations which preserve the multiplicity of eigenvalues