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

We consider the eigenvalue problem for the fractional Laplacian in a bounded domain 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, 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 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