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

We have previously discussed the one dimensional multibarrier potential of finite range and found that under certain values of its total length L and the ratio c of total interval to total width it shows signs of chaos as well as phase transition and a transmission amplitude of unity. We show here that, like the infinite Kronig-Penney system which is arrayed along the whole spatial region it also have a band-gap structure associated with its energy spectrum. But unlike the Kronig-Penney system in which the gaps disappear for large values of the energies here these gaps, because of their dependence upon L and c, do not diminish for certain values of these parameters. We, also, show that the energy spectrum is discontinuous at many of its points even at the parts in which it has no gaps. We show that these results, together with others, signify that for these values of L and c the energy spactrum of the bounded large multibarrier system is singular. PACS numbers: 71.15.Ap, 02.10.Yn, 03.65.Nk, 02.40.Xx