  We present a study of the one-particle spectral properties for a variety of
models of Luttinger liquids with open boundaries. We first consider the
Tomonaga-Luttinger model using bosonization. For weak interactions the boundary
exponent of the power-law suppression of the spectral weight close to the
chemical potential is dominated by a term linear in the interaction. This
motivates us to study the spectral properties also within the Hartree-Fock
approximation. It already gives power-law behavior and qualitative agreement
with the exact spectral function. For the lattice model of spinless fermions
and the Hubbard model we present numerically exact results obtained using the
density-matrix renormalization-group algorithm. We show that many aspects of
the behavior of the spectral function close to the boundary can again be
understood within the Hartree-Fock approximation. For the repulsive Hubbard
model with interaction U the spectral weight is enhanced in a large energy
range around the chemical potential. At smaller energies a power-law
suppression, as predicted by bosonization, sets in. We present an analytical
discussion of the crossover and show that for small U it occurs at energies
exponentially (in -1/U) close to the chemical potential, i.e. that bosonization
only holds on exponentially small energy scales. We show that such a crossover
can also be found in other models.
