Super periodic potential

Highlights: • Concept of Super Periodic (SPP) potential of arbitrary order is introduced. • Scattering coefficients for arbitrary SPP are calculated in closed form. • Symmetric Fractal potential is derived as a special case of SPP. • Generalized Cantor and Smith–Volterra–Cantor (SVC) potentials are studied as special cases of SPP. • Tunnelling amplitudes for fractal, Cantor and SVC potentials are derived. - 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.