  We investigate the spectral function of Bloch states in an one-dimensional
tight-binding non-interacting chain with two different models of static
correlated disorder, at zero temperature. We report numerical calculations of
the single-particle spectral function based on the Kernel Polynomial Method,
which has an $\mathcal{O}(N)$ computational complexity. These results are then
confirmed by analytical calculations, where precise conditions were obtained
for the appearance of a classical limit in a single-band lattice system.
Spatial correlations in the disordered potential give rise to non-perturbative
spectral functions shaped as the probability distribution of the random on-site
energies, even at low disorder strengths. In the case of disordered potentials
with an algebraic power-spectrum, $\propto\left|k\right|^{-\alpha}$, we show
that the spectral function is not self-averaging for $\alpha\geq1$.
