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--Kwaśnicki--Małecki--Stós and Kwaśnicki, and confirms the conjectural Oα(n−2) remainder suggested by the numerical simulations of Kaleta--Kwaśnicki--Małecki. Moreover, we 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. 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.
Related
- Corner Regularity of Laplacian Eigenfunctions
- Cherednik integrable system: eigenfunctions at generic eigenvalues
- Positive, monotone, and convex Gaussian process regression with Laplacian eigenfunctions
- Harmonic-Einstein maps and critical metrics for eigenvalues of the conformal Laplacian
- Arithmetic classification of the Dirichlet eigenvalues for a one-dimensional nonlocal p-Laplacian problem
BibTeX
@misc{zhang2026eigenvalues,
title = {{Eigenvalues and eigenfunctions of the fractional Laplacian on the interval}},
author = {Cheng Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.23457},
eprint = {2608.23457},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.23457},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Eigenvalues and eigenfunctions of the fractional Laplacian on the interval
AU - Cheng Zhang
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2608.23457
UR - https://arxiv.org/abs/2608.23457
AB - 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--Kwaśnicki--Małecki--Stós and Kwaśnicki, and confirms the conjectural O_α(n^−2) remainder suggested by the numerical simulations of Kaleta--Kwaśnicki--Małecki. Moreover, we 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. 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.
ER -