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We use analytic and numerical methods to determine the density of states of aone-dimensional electron gas coupled to a spatially random quasi-staticback-scattering potential of long correlation length. Our results provideinsight into the 'pseudogap' phenomenon occurring in underdoped high-Tcsuperconductors, quasi-one-dimensional organic conductors and liquid metals.They demonstrate the important role played by amplitude fluctuations of thebackscattering potential and by fluctuations in gradients of the potential, andconfirm the importance of the self-consistency which is a key feature of the'FLEX'-type approximations for the electron Green's function. Our results allowan assessment of the merits of different approximations: a previous approximatetreatment presented by Sadovskii and, we show, justified by a WKB approximationgives a reasonably good representation, except for a ``central peak'' anomaly,of our numerically computed densities of states, whereas a previousapproximation introduced by Lee, Rice and Anderson is not as accurate.Comment: 19 pages, 5 figures include

Pseudogaps in one-dimensional models with quasi-long-range order