We develop a simple variational approximation for the ground state of
a conduction band in the presence of a localized core-hole potential,
which fulfils the Anderson orthogonality theorem. Using this as the in
itial state of the Auger decay, we derive an expression for the line s
hape, including correlation effects in the Ladder Approximation. The c
omputed spectrum is a combination of one-hole and two-hole-one-electro
n contributions; it is similar to the experimental spectra of low -ele
ctron-density systems that show the apparent negative U behaviour. The
se results show that in such systems the conduction electrons effectiv
ely screen the core hole before it decays. Also, we present evidence f
or a transition from complete relaxation to no relaxation with increas
ing band filling; the ''unrelaxed'' contribution to the spectra reflec
ts a two-holes density of states.