  In the spinless Holstein model at half-filling the coupling of electrons to
phonons is responsible for a phase transition from a metallic state at small
coupling to a Peierls distorted insulated state when the electron-phonon
coupling exceeds a critical value. For the adiabatic case of small phonon
frequencies, the transition is accompanied by a phonon softening at the
Brillouin zone boundary whereas a hardening of the phonon mode occurs in the
anti-adiabatic case. The phonon dynamics studied in this letter do not only
reveal the expected renormalization of the phonon modes but also show
remarkable additional contributions due to electronic particle-hole
excitations.
