Eigenvalues of the fractional Laplace operator in the unit ball

TL;DR

This paper introduces an efficient numerical scheme to bound eigenvalues of the fractional Laplace operator in the unit ball, using variational and intermediate methods with orthogonal polynomials, partially resolving a conjecture about eigenfunction symmetry.

Contribution

It develops a novel numerical approach combining Rayleigh-Ritz and Aronszajn methods with explicit polynomial expressions to estimate eigenvalues of the fractional Laplacian.

Findings

Proves the second eigenvalue corresponds to an antisymmetric function for certain dimensions and alpha values.

Provides numerical evidence supporting the conjecture for various dimensions and alpha in (0,2].

Offers a partially analytical resolution to a conjecture on eigenfunction symmetry.

Abstract

We describe a highly efficient numerical scheme for finding two-sided bounds for the eigenvalues of the fractional Laplace operator (-Delta)^{alpha/2} in the unit ball D in R^d, with a Dirichlet condition in the complement of D. The standard Rayleigh-Ritz variational method is used for the upper bounds, while the lower bounds involve the less-known Aronszajn method of intermediate problems. Both require explicit expressions for the fractional Laplace operator applied to a linearly dense set of functions in L^2(D). We use appropriate Jacobi-type orthogonal polynomials, which were studied in a companion paper "Fractional Laplace operator and Meijer G-function". Our numerical scheme can be applied analytically when polynomials of degree two are involved. This is used to partially resolve the conjecture of T. Kulczycki, which claims that the second smallest eigenvalue corresponds to an antisymmetric function: we prove that this is the case when either d <= 2 and 0 < alpha <= 2, or d <= 9 and alpha = 1, and we provide strong numerical evidence for d <= 9 and general alpha in (0,2].