We calculate the damping γ 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 /2m at zero temperature. Using standard many-body perturbation theory, we obtain γ q from the expansion of the inverse irreducible polarization to first order in the effective screened (RPA) interaction. For wave-vectors |q|/k F ≪ F (where k F = mv F is the Fermi wave-vector) we find to leading order γ q ∝ |q| 3 /(v F m 2 ). On the other hand, for F ≪ |q|/k F most of the spectral weight is carried by the particle-hole continuum, which is distributed over a frequency interval of the order of q 2 /m. We also show that zero sound damping leads to a finite maximum proportional to |k − k F | −2+2η of the charge peak in the single-particle spectral function, where η is the anomalous dimension. Our prediction agrees with photoemission data for the blue bronze K 0.3 MoO 3 . We comment on other recent calculations of γ q .

We calculate the dynamic structure factor S(ω, q) of spinless fermions in one dimension with quadratic energy dispersion k 2 /2m and long range density-density interaction whose Fourier transform fq is dominated by small momentum-transfers q q0 ≪ kF . Here q0 is a momentum-transfer cutoff and kF is the Fermi momentum. Using functional bosonization and the known properties of symmetrized closed fermion loops, we obtain an expansion of the inverse irreducible polarization to second order in the small parameter q0/kF . In contrast to perturbation theory based on conventional bosonization, our functional bosonization approach is not plagued by mass-shell singularities. For interactions which can be expanded as fq = f0 + f ′′ 0 q 2 /2 + O(q 4 ) with f ′′ 0 = 0 we show that the momentum scale qc = 1/|mf ′′ 0 | separates two regimes characterized by a different q-dependence of the width γq of the collective zero sound mode and other features of S(ω, q). For qc ≪ q ≪ kF all integrations in our functional bosonization result for S(ω, q) can be evaluated analytically; we find that the line-shape in this regime is non-Lorentzian with an overall width γq ∝ q 3 /(mqc) and a threshold singularitywhere v is the velocity of the zero sound mode. Assuming that higher orders in perturbation theory transform the logarithmic singularity into an algebraic one, we find for the corresponding threshold exponent µq = 1 − 2ηq with ηq ∝ q 2 c /q 2 . Although for q qc we have not succeeded to explicitly evaluate our functional bosonization result for S(ω, q), we argue that for any one-dimensional model belonging to the Luttinger liquid universality class the width of the zero sound mode scales as q 2 /m for q → 0.