  We consider a single atom in an optical lattice, subject to a harmonic
trapping potential. The problem is treated in the tight-binding approximation,
with an extra parameter \kappa denoting the strength of the harmonic trap. It
is shown that the \kappa \to 0 limit of this problem is singular, in the sense
that the density of states for a very shallow trap (\kappa \to 0) is {\it
qualitatively different} from that of a translationally invariant lattice
(\kappa = 0). The physics of this difference is discussed, and densities of
states and wave functions are exhibited and explained.
