A note on the persistence of multiplicity of eigenvalues of fractional Laplacian under perturbations

We consider the eigenvalue problem for the fractional Laplacian (-Delta)(s), s is an element of (0, 1), in a bounded domain Omega with Dirichlet boundary condition. A recent result (see Fall et al., 2023) states that, under generic small perturbations of the coefficient of the equation or of the domain Omega , all the eigenvalues are simple. In this paper we give a condition for which a perturbation of the coefficient or of the domain preserves the multiplicity of a given eigenvalue. Also, in the case of an eigenvalue of multiplicity v = 2 we prove that the set of perturbations of the coefficients which preserve the multiplicity is a smooth manifold of codimension 2 in C-1(R-n) .