Super Periodic Potential

TL;DR

This paper introduces the concept of super periodic potential (SPP) in one dimension, providing a general wave propagation theory, analytical reflection and transmission coefficients, and applications to fractal potentials like Cantor and Smith-Volterra-Cantor.

Contribution

It develops a comprehensive formalism for SPP of arbitrary order, deriving closed-form expressions for scattering and tunneling in complex fractal potentials.

Findings

Closed-form reflection and transmission coefficients for SPP.

Analytical tunneling amplitudes for fractal potentials.

Identification of symmetric self-similarity as a special case of super periodicity.

Abstract

In this paper we introduce the concept of super periodic potential (SPP) of arbitrary order $n$, $n \in I^{+}$, in one dimension. General theory of wave propagation through SPP of order $n$ is presented and the reflection and transmission coefficients are derived in their closed analytical form by transfer matrix formulation. We present scattering features of super periodic rectangular potential and super periodic delta potential as special cases of SPP. It is found that the symmetric self-similarity is the special case of super periodicity. Thus by identifying a symmetric fractal potential as special cases of SPP, one can obtain the tunnelling amplitude for a particle from such fractal potential. By using the formalism of SPP we obtain the close form expression of tunnelling amplitude of a particle for general Cantor and Smith-Volterra-Cantor potentials.