Fractional Laplacian in Bounded Domains

TL;DR

This paper develops a discretized fractional Laplacian suitable for bounded domains, incorporating boundary conditions inspired by physical models, and analyzes its eigenvalues and eigenfunctions.

Contribution

It introduces a novel discretization method for the fractional Laplacian in finite intervals with boundary conditions, supported by physical model justifications.

Findings

Eigenvalues and eigenfunctions computed numerically for various boundary conditions.

Analytical insights into the eigenvalue spectrum structure.

Implementation validated through physical models like hopping particles and elastic springs.

Abstract

The fractional Laplacian operator, $-(-\triangle)^{\frac{\alpha}{2}}$, appears in a wide class of physical systems, including L\'evy flights and stochastic interfaces. In this paper, we provide a discretized version of this operator which is well suited to deal with boundary conditions on a finite interval. The implementation of boundary conditions is justified by appealing to two physical models, namely hopping particles and elastic springs. The eigenvalues and eigenfunctions in a bounded domain are then obtained numerically for different boundary conditions. Some analytical results concerning the structure of the eigenvalues spectrum are also obtained.