Eigenvalues and eigenfunctions of the fractional Laplacian on the interval

Abstract

We prove a three-term asymptotic formula for the eigenvalues of the fractional Laplacian on the bounded interval $(-1,1)$. This improves the eigenvalue asymptotics of Kulczycki--Kwa\'snicki--Ma{\l}ecki--St\'os and Kwa\'snicki, and confirms the conjectural $O_\alpha(n^{-2})$ remainder suggested by the numerical simulations of Kaleta--Kwa\'snicki--Ma{\l}ecki. Moreover, we prove that the normalized eigenfunctions are bounded uniformly in the eigenvalue index $n$ and the fractional order $\alpha$. This settles the conjecture proposed by Kwa\'snicki through numerical experiments. Furthermore, we prove that the $n$-th eigenfunction has exactly $n-1$ zeros in the interval $(-1,1)$ and every zero is simple, and hence there are exactly $n$ nodal domains. A key ingredient in the proof is an explicit representation of the eigenfunction.