  The spinless Falicov-Kimball model is studied by the use of a recently
developed projector-based renormalization method (PRM) for many-particle
Hamiltonians. The method is used to evaluate static and dynamic quantities of
the one-dimensional model at half-filling. To these belong the quasiparticle
excitation energy $\tilde{\epsilon}_k$ and the momentum distribution $n_k$ of
the conduction electrons and spatial correlation functions of the localized
electrons. One of the most remarkable results is the appearance of a gap in
$\tilde{\epsilon}_k$ at the Fermi level of the order of the Coulomb repulsion
$U$, which is accompanied by a smooth behavior for $n_k$. The density of states
for the conduction electrons and the one-particle spectral functions for the
localized electrons are also discussed. In both quantities a gap opens with
increasing $U$.
