This article studies the eigenvalue problem of a fractional differential equation which is a foundation model of a bar of finite length with long-range interactions arising from non-local continuum mechanics. We show that this problem has countable simple real eigenvalues and the corresponding eigenfunctions form a complete orthogonal system in the Hilbert space L2. Furthermore, the asymptotic behavior of eigenvalues and the numbers of zeros of eigenfunctions are studied by using the analytic perturbation theory.

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