  We study the spectral function of interacting one-dimensional fermions for an
integrable lattice model away from half-filling. The divergent power-law
singularity of the spectral function near the single-particle or single-hole
energy is described by an effective x-ray edge type model. At low densities and
for momentum near the zone boundary, we find a second divergent singularity at
higher energies which is associated with a two-particle bound state. We use the
Bethe ansatz solution of the model to calculate the exact singularity exponents
for any momentum and for arbitrary values of chemical potential and interaction
strength in the critical regime. We relate the singularities of the spectral
function to the long-time decay of the fermion Green's function and compare our
predictions with numerical results from the time-dependent density matrix
renormalization group (tDMRG). Our results show that the tDMRG method is able
to provide accurate time decay exponents in the cases of power-law decay of the
Green's function. Some implications for the line shape of the dynamical
structure factor away from half-filling are also discussed. In addition, the
spectral weight of the Luttinger liquid result for the dynamical structure
factor of the Heisenberg model at zero field is compared with the exact
two-spinon contribution.
