  We show that the concept of umklapp-scattering driven instabilities in
one-dimensional systems can be generalized to arbitrary multiple
umklapp-scattering processes at commensurate fillings given that the system has
sufficiently longer range interactions. To this end we study the fundamental
model system, namely interacting spinless fermions on a one-dimensional
lattice, via a density matrix renormalization group approach. The instabilities
are investigated via a new method allowing to calculate the ground-state charge
stiffness numerically exactly. The method can be used to determine other ground
state susceptibilities in general.
