Band structures in transmission coefficients generated by Dirac comb potentials

Using the threshold conditions and bound state energies investigated earlier by us as a critical input we systematically study the nature of band formation in the transmission coefficient generated by Dirac comb potentials having equispaced (i) attractive, (ii) repulsive and (iii) alternating attractive and repulsive delta terms having same strength and confined within a fixed range. We find that positions of the peaks of transmission coefficient generated by a combination of one attractive and one repulsive delta terms having same strength and separated by gap a is independent of the potential strength and coincide with the energy eigenvalues of 1D box of range a. We further study analytically and numerically the transmission across Dirac comb potentials containing two or three delta terms and these results are useful in the analysis of the transmission in the general case. In the case of Dirac comb potentials containing Na attractive delta terms we find that the nature of the first band and higher bands of the transmission coefficient are different, and if such a potential generates Nb number of bound states, the first band in the transmission coefficient generated by the potential has NT1=Na−NbNT1=Na−Nb peaks. In the case of higher bands generated by delta comb potential having N   delta terms each band has N−1N−1 peaks. Further we systematically study the behavior of band gaps and band spread as a function of potential strength and number of terms in the Dirac comb. The results obtained by us provide a relation between bound state spectrum, number of delta terms in the Dirac comb and the band pattern which can be explored for potential applications.