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We consider the eigenvalue problem for the restricted fractional Laplacian ina bounded domain with homogeneous Dirichlet boundary conditions. We introducethe notion of fractional capacity for compact subsets, with the property thatthe eigenvalues are not affected by the removal of zero fractional capacitysets. Given a simple eigenvalue, we remove from the domain a family of compactsets which are concentrating to a set of zero fractional capacity and we detectthe asymptotic expansion of the eigenvalue variation; this expansion depends onthe eigenfunction associated to the limit eigenvalue. Finally, we study thecase in which the family of compact sets is concentrating to a point.

On simple eigenvalues of the fractional Laplacian under removal of small fractional capacity sets