Eigenvalues of the fractional Laplace operator in the interval

Two-term Weyl-type asymptotic law for the eigenvalues of the one-dimensional fractional Laplace operator (−Δ)α/2 (α∈(0,2)) in the interval (−1,1) is given: the n-th eigenvalue is equal to (nπ/2−α(2−α)π/8)+O(1/n). Simplicity of eigenvalues is proved for α∈[1,2). L2 and L∞ properties of eigenfunctions are studied. We also give precise numerical bounds for the first few eigenvalues.