  We calculate the damping gamma_q of collective density oscillations (zero
sound) in a one-dimensional Fermi gas with dimensionless forward scattering
interaction F and quadratic energy dispersion k^2 / 2 m at zero temperature.
For wave-vectors | q| /k_F small compared with F we find to leading order
gamma_q = v_F^{-1} m^{-2} Y (F) | q |^3, where v_F is the Fermi velocity, k_F
is the Fermi wave-vector, and Y (F) is proportional to F^3 for small F. We also
show that zero-sound damping leads to a finite maximum proportional to |k - k_F
|^{-2 + 2 eta} of the charge peak in the single-particle spectral function,
where eta is the anomalous dimension. Our prediction agrees with photoemission
data for the blue bronze K_{0.3}MoO_3.
