Eigenvalues of the fractional Laplace operator in the interval

TL;DR

This paper derives asymptotic formulas for the eigenvalues of the one-dimensional fractional Laplace operator in an interval, proves eigenvalue simplicity for certain alpha, and studies eigenfunction properties with numerical bounds.

Contribution

It provides a two-term Weyl-type asymptotic law for fractional Laplace eigenvalues and establishes eigenvalue simplicity for alpha in [1, 2).

Findings

Asymptotic formula for eigenvalues: (n pi/2 - (2 - alpha) pi/8)^alpha + O(1/n)

Eigenvalues are simple for alpha in [1, 2)

Numerical bounds for the first few eigenvalues are provided

Abstract

Two-term Weyl-type asymptotic law for the eigenvalues of one-dimensional fractional Laplace operator (-d^2/dx^2)^(alpha/2) (0 < alpha < 2) in the interval (-1,1) is given: the n-th eigenvalue is equal to (n pi/2 - (2 - alpha) pi/8)^alpha + O(1/n). Simplicity of eigenvalues is proved for alpha in [1, 2). L^2 and L^infinity properties of eigenfunctions are studied. We also give precise numerical bounds for the first few eigenvalues.