Single-atom density of states of an optical lattice

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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-->0 limit of this problem is singular,

in the sense that the density of states for a very shallow trap (kappa-->0)

is 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.