Quantum tunneling from a new type of generalized Smith-Volterra-Cantor potential

Abstract

In this paper, we introduce and analyze the Smith-Volterra-Cantor potential of power \( n \), denoted as SVC\(\left(ρ, n\right)\). Bridging the gap between the general Cantor and SVC systems, this novel potential offers a fresh perspective on Cantor-like potential systems within quantum mechanics that unify fractal and non-fractal potentials. Utilizing the Super Periodic Potential (SPP) formalism, we derive the close form expression of the transmission probability \( T_{G}(k) \). Notably, the system exhibits exceptionally sharp transmission resonances, a characteristic that distinguishes it from other quantum systems. Furthermore, the multifaceted transmission attributes of the SVC\(\left(ρ, n\right)\) are found to be critically dependent on both parameters, \( ρ\) and \( n \), offering an intricate interplay that warrants deeper exploration. Our findings highlight a pronounced scaling behavior of reflection probability with \( k \), which is underpinned by analytical derivations.

25 Pages, 12 figures