  We use analytic and numerical methods to determine the density of states of a
one-dimensional electron gas coupled to a spatially random quasi-static
back-scattering potential of long correlation length. Our results provide
insight into the 'pseudogap' phenomenon occurring in underdoped high-Tc
superconductors, quasi-one-dimensional organic conductors and liquid metals.
They demonstrate the important role played by amplitude fluctuations of the
backscattering potential and by fluctuations in gradients of the potential, and
confirm the importance of the self-consistency which is a key feature of the
'FLEX'-type approximations for the electron Green's function. Our results allow
an assessment of the merits of different approximations: a previous approximate
treatment presented by Sadovskii and, we show, justified by a WKB approximation
gives a reasonably good representation, except for a ``central peak'' anomaly,
of our numerically computed densities of states, whereas a previous
approximation introduced by Lee, Rice and Anderson is not as accurate.
