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--Kwaśnicki--Małecki--Stós and Kwaśnicki, and confirms the conjectural $O_α(n<sup>{-2})$ remainder by the numerical simulations of Kaleta--Kwaśnicki--Małecki. We also prove that the normalized eigenfunctions are bounded uniformly in the eigenvalue index $n$ and the fractional order $α$. This settles the conjecture proposed by Kwaśnicki through numerical experiments.