  Including the previously ignored dispersion of phonons we revisit the
metal-insulator transition problem in one-dimensional electron-phonon systems
on the basis of a modified spinless fermion Holstein model. Using
matrix-product-state techniques we determine the global ground-state phase
diagram in the thermodynamic limit for the one-dimensional band case, and show
that in particular the curvature of the bare phonon band has a significant
effect, not only on the transport properties characterized by the conductance
and the Luttinger liquid parameter, but also on the phase space structure of
the model as a whole. While a downward curved (convex) dispersion of the
phonons only shifts the Tomonaga-Luttinger-liquid to charge-density-wave
quantum phase transition towards stronger EP coupling, an upward curved
(concave) phonon band leads to a new phase-separated state which, in the case
of strong dispersion, can even completely cover the charge-density wave. Such
phase separation does not occur in the related Edwards fermion-boson model.
