Fractional Laplace operator and Meijer G-function

TL;DR

This paper extends the class of functions explicitly computable under the fractional Laplace operator, notably involving Meijer G-functions and hypergeometric functions, and constructs eigenfunctions for a related boundary value problem.

Contribution

It demonstrates that the fractional Laplace operator maps certain Meijer G-functions and hypergeometric functions into the same class, and constructs a complete system of eigenfunctions for a boundary value problem.

Findings

Explicit computation of fractional Laplace operator on Meijer G-functions.

Construction of eigenfunctions for fractional Laplacian with Dirichlet boundary conditions.

Foundation for estimating eigenvalues in a subsequent study.

Abstract

We significantly expand the number of functions whose image under the fractional Laplace operator can be computed explicitly. In particular, we show that the fractional Laplace operator maps Meijer G-functions of |x|^2, or generalized hypergeometric functions of -|x|^2, multiplied by a solid harmonic polynomial, into the same class of functions. As one important application of this result, we produce a complete system of eigenfunctions of the operator (1-|x|^2)_+^{alpha/2} (-Delta)^{alpha/2} with the Dirichlet boundary conditions outside of the unit ball. The latter result will be used to estimate the eigenvalues of the fractional Laplace operator in the unit ball in a companion paper "Eigenvalues of the fractional Laplace operator in the unit ball".