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 (− )α/2 (α ∈ (0, 2)) in the interval (−1, 1) is given: the n-th eigenvalue is equal to (nπ/2 − (2 − α)π/8)α + 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. © 2011 Elsevier Inc. All rights reserved