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\(\left(\rho, n\right)\). 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\(\left(\rho, n\right)\) are found to be critically dependent on both parameters, \( \rho \) 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.