L\'evy Transport in Slab Geometry of Inhomogeneous Media

TL;DR

This paper investigates fractional quantum dynamics in complex media, deriving an effective fractional Schrödinger equation that models wave transport with Le9vy flights, supported by numerical and analytical results.

Contribution

It introduces a novel effective fractional Schrf6dinger equation for wave transport in inhomogeneous media, linking fractional calculus with quantum dynamics.

Findings

Derived an effective fractional Schrf6dinger equation for complex media

Numerically demonstrated Le9vy flights in an infinite potential well

Obtained analytical expression for the effective wave function

Abstract

We present a physical example, where a fractional (both in space and time) Schr\"odinger equation appears only as a formal effective description of diffusive wave transport in complex inhomogeneous media. This description is a result of the parabolic equation approximation that corresponds to the paraxial small angle approximation of the fractional Helmholtz equation. The obtained effective quantum dynamics is fractional in both space and time. As an example, L\'evy flights in an infinite potential well are considered numerically. An analytical expression for the effective wave function of the quantum dynamics is obtained as well.