  We investigate the influence of the electron-phonon coupling in the
one-dimensional spinless Holstein model at half-filling using both a recently
developed projector-based renormalization method (PRM) and an refined exact
diagonalization technique in combination with the kernel polynomial method. At
finite phonon frequencies the system shows a metal-insulator transition
accompanied by the appearance of a Peierls distorted state at a finite critical
electron-phonon coupling. We analyze the opening of a gap in terms of the
(inverse) photoemission spectral functions which are evaluated in both
approaches. Moreover, the PRM approach reveals the softening of a phonon at the
Brillouin-zone boundary which can be understood as precursor effect of the gap
formation.
