Generic properties of eigenvalues of the fractional Laplacian

Abstract

We consider the Dirichlet eigenvalues of the fractional Laplacian $(-\Delta {)}^s$, with $s\in (0,1)$, related to a smooth bounded domain $\Omega$. We prove that there exists an arbitrarily small perturbation $\tilde{\Omega }=(I+\psi )(\Omega )$ of the original domain such that all Dirichlet eigenvalues of the fractional Laplacian associated to $\tilde{\Omega }$ 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.