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

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 $O_{\alpha }(n^{-2})$ 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 $\alpha$. This settles the conjecture proposed by Kwaśnicki through numerical experiments.