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

We consider the eigenvalue problem for the fractional Laplacian ${(-\Delta )}^s$, $s\in (0,1)$, 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 $\nu =2$ we prove that the set of perturbations of the coefficients which preserve the multiplicity is a smooth manifold of codimension 2 in $C^1\left(R^n\right)$.