Microscopic versus mesoscopic local density of states in one-dimensional localization

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We calculate the probability distribution of the local density of states ν in a disordered one-dimensional conductor or single-mode waveguide, attached at one end to an electron or photon reservoir. We show that this distribution does not display a log-normal tail for small ν, but diverges instead ∝ν⁻¹/². The log-normal tail appears if ν is averaged over rapid oscillations on the scale of the wavelength. There is no such qualitative distinction between microscopic and mesoscopic densities of states if the levels are broadened by inelastic scattering or absorption, rather than by coupling to a reservoir.