Boundary condition effects on the energy spectrum of an one dimensional imperfect crystal*

We report the energy spectrum and the ground density profile of an interactionless particle gas in an one-dimensional crystal with imperfections. The imperfect crystal is modeled by a Dirac comb potential, where some delta strengths are different from the rest.
We calculate the energy spectrum using the transfer matrix-method for periodic and Dirichlet boundary conditions. We find that the spectrum obtained with periodic conditions is more general since when the strength of a delta becomes infinite we recover the Dirichlet spectrum. The density profile becomes localized around the crystal vacancies.
We observe the emergence of an energy gap between the ground and the first excited states that promotes a Bose-Einstein condensation in a interactionless Bose gas within the imperfect 1D crystal. The energy spectra with both boundary conditions are obtained and compared for several delta strengths.