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 (-Delta)(alpha/2) (alpha is an element of (0, 2)) in the interval (-1, 1) is given: the n-th eigenvalue is equal to (n pi/2 - (2 - alpha)pi/8)(alpha) O(1/n). Simplicity of eigenvalues is proved for alpha is an element of [1, 2). L-2 and L-infinity properties of eigenfunctions are studied. We also give precise numerical bounds for the first few eigenvalues. (C) 2011 Elsevier Inc. All rights reserved.