  All Bloch states of the mean field of a Bose-Einstein condensate in the
presence of a one dimensional lattice of impurities are presented in closed
analytic form. The band structure is investigated by analyzing the stationary
states of the nonlinear Schr\"odinger, or Gross-Pitaevskii, equation for both
repulsive and attractive condensates. The appearance of swallowtails in the
bands is examined and interpreted in terms of the condensates superfluid
properties. The nonlinear stability properties of the Bloch states are
described and the stable regions of the bands and swallowtails are mapped out.
We find that the Kronig-Penney potential has the same properties as a
sinusoidal potential; Bose-Einstein condensates are trapped in sinusoidal
optical lattices. The Kronig-Penney potential has the advantage of being
analytically tractable, unlike the sinusoidal potential, and, therefore, serves
as a good model for experimental phenomena.
