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 κdenoting the strength of the harmonic trap. It is shown that the κ→ 0 limit of this problem is singular, in the sense that the density of states for a very shallow trap (κ→ 0) is qualitatively different from that of a translationally invariant lattice (κ= 0). The physics of this difference is discussed, and densities of states and wave functions are exhibited and explained.