The band-gap structure and the singular character of the bounded large array of potential barriers

Abstract

The bounded one dimensional multibarrier potential shows signs of chaos, phase transition and a transmission probability of unity for certain values of its total length $L$ and the ratio $c$ of total interval to total width. Like the infinite Kronig-Penney system, which is arranged along the whole spatial region, the bounded multibarrier potential has a band-gap structure in its energy spectrum. But unlike the Kronig-Penney system, in which the gaps disappear for large energies, these gaps do not disappear for certain values of $L$ and $c$. The energy is discontinuous even in parts of the spectrum with no gaps at all. These results imply that the energy spectrum of the bounded multibarrier system is singular.