Quantum tunneling from family of Cantor potentials in fractional quantum mechanics

TL;DR

This paper investigates quantum tunneling in space fractional quantum mechanics using Cantor and Smith-Volterra-Cantor potentials, revealing new scattering features like energy bands and sharp transmission peaks.

Contribution

First study of quantum tunneling through fractal potentials in fractional quantum mechanics, uncovering novel scattering phenomena and analytical scaling relations.

Findings

Emergence of energy-band like features

Sharp transmission resonances

Analytical scaling relation of scattering amplitude

Abstract

We explore the features of non-relativistic quantum tunneling in space fractional quantum mechanics through a family of Cantor potentials. We consider two types of potentials: general Cantor and general Smith-Volterra-Cantor potential. The Cantor potential is an example of fractal potential while the Smith-Volterra-Cantor potential doesn't belong to the category of a fractal system. The present study brings for the first time, the study of quantum tunneling through fractal potential in fractional quantum mechanics. We report several new features of scattering in the domain of space fractional quantum mechanics including the emergence of energy-band like features from these systems and extremely sharp transmission features. Further the scaling relation of the scattering amplitude with wave vector $k$ is presented analytically for both types of potentials.