Definition of the Riesz Derivative and its Application to Space Fractional Quantum Mechanics

TL;DR

This paper explores various mathematical representations of the Riesz derivative, emphasizing its role in space fractional quantum mechanics and analyzing the behavior as the order approaches one.

Contribution

It compares different definitions of the Riesz derivative and examines the alpha equals one limit in space fractional quantum mechanics for consistency.

Findings

Different representations of the Riesz derivative are equivalent at alpha=1.

The behavior of space fractional quantum mechanics as alpha approaches 1 is consistent.

The Fourier transform-based definition of the Riesz derivative is validated.

Abstract

We investigate and compare different representations of the Riesz derivative, which plays an important role in anomalous diffusion and space fractional quantum mechanics. In particular, we show that a certain representation of the Riesz derivative that is generally given as also valid for order alpha equals 1, behaves no differently than the other definition given in terms of its Fourier transform. In the light of this, we discuss the alpha goes to 1 limit of the space fractional quantum mechanics and its consistency.