Polyadic Cantor potential of minimum lacunarity: Special case of super periodic generalized unified Cantor potential

TL;DR

This paper introduces a generalized Cantor potential framework incorporating super periodic structures, deriving analytical expressions for quantum tunneling and revealing unique transmission resonance behaviors.

Contribution

It presents a new generalized unified Cantor potential model with super periodic formalism, providing closed-form transmission probability expressions and analyzing their quantum tunneling properties.

Findings

GUCP exhibits sharp transmission resonances.

Transmission saturation occurs with increasing stages.

Scaling laws relate reflection probability to wave vector.

Abstract

To bridge the fractal and non-fractal potentials we introduce the concept of generalized unified Cantor potential (GUCP) with the key parameter $N$ which represents the potential count at the stage $S=1$. This system is characterized by total span $L$, stages $S$, scaling parameter $\rho$ and two real numbers $\mu$ and $\nu$. Notably, the polyadic Cantor potential (PCP) system with minimal lacunarity is a specific instance within the GUCP paradigm. Employing the super periodic potential (SPP) formalism, we formulated a closed-form expression for transmission probability $T_{S}(k, N)$ using the $q$-Pochhammer symbol and investigated the features of non-relativistic quantum tunneling through this potential configuration. We show that GUCP system exhibits sharp transmission resonances, differing from traditional quantum systems. Our analysis reveals saturation in the transmission profile with evolving stages $S$ and establishes a significant scaling relationship between reflection probability and wave vector $k$ through analytical derivations.