Eigenvalue asymptotics and uniform eigenfunction bounds for the fractional Laplacian in the interval

Abstract

We prove a three-term asymptotic formula for the eigenvalues of the fractional Laplacian in the bounded interval. 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 by the numerical simulations of Kaleta--Kwa\'snicki--Ma{\l}ecki. We also 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.