Simple Case of Fractional Sturm-Liouville Problem with Homogeneous von Neumann Boundary Conditions

We study a variant of fractional Sturm-Liouvile eigenvalue problem with homogeneous von Neumann boundary conditions and prove that its spectrum is purely discrete. The differential fractional eigenvalue problem is converted to the integral one determined by the compact, self-adjoint Hilbert-Schmidt integral operator. Both eigenvalue problems, differential and integral one, are equivalent on the respective subspace of continuous functions. The eigenfunctions are continuous and form an orthogonal basis in the respective Hilbert space.