Generic properties of eigenvalues of the fractional Laplacian

We consider the Dirichlet eigenvalues of the fractional Laplacian $(-\mathrm{\Delta }{)}^s$, with $s\in (0,1)$, related to a smooth bounded domain $\mathrm{\Omega }$. We prove that there exists an arbitrarily small perturbation $\tilde{\mathrm{\Omega }}=(I+\psi )(\mathrm{\Omega })$ of the original domain such that all Dirichlet eigenvalues of the fractional Laplacian associated to $\tilde{\mathrm{\Omega }}$ are simple. As a consequence we obtain that all Dirichlet eigenvalues of the fractional Laplacian on an interval are simple. In addition, we prove that for a generic choice of parameters all the eigenvalues of some non-local operators are also simple.