Fourier transforms, fractional derivatives, and a little bit of quantum mechanics

TL;DR

This paper explores the mathematical properties of Fourier-based fractional derivatives, extending their application to tempered distributions and demonstrating their relevance in quantum mechanics.

Contribution

It introduces a Fourier transform-based fractional derivative, analyzes its properties on test functions and distributions, and applies it to quantum mechanics contexts.

Findings

Fractional derivatives defined via Fourier transforms have well-defined actions on test functions.

The extension to tempered distributions broadens the applicability of fractional derivatives.

Application to quantum mechanics illustrates the practical relevance of the mathematical framework.

Abstract

We discuss some of the mathematical properties of the fractional derivative defined by means of Fourier transforms. We first consider its action on the set of test functions $\Sc(\mathbb R)$, and then we extend it to its dual set, $\Sc'(\mathbb R)$, the set of tempered distributions, provided they satisfy some mild conditions. We discuss some examples, and we show how our definition can be used in a quantum mechanical context.