  Motivated by recent scanning tunneling and photoemission spectroscopy
measurements on self-organized gold chains on a germanium surface we
reinvestigate the local single-particle spectral properties of Luttinger
liquids. In the first part we use the bosonization approach to exactly compute
the local spectral function of a simplified field theoretical low-energy model
and take a closer look at scaling properties as a function of the ratio of
energy and temperature. Translational invariant Luttinger liquids as well as
those with an open boundary (cut chain geometry) are considered. We explicitly
show that the scaling functions of both setups have the same analytic form. The
scaling behavior suggests a variety of consistency checks which can be
performed on measured data to experimentally verify Luttinger liquid behavior.
In a second part we approximately compute the local spectral function of a
microscopic lattice model---the extended Hubbard model---close to an open
boundary using the functional renormalization group. We show that as a function
of energy and temperature it follows the field theoretical prediction in the
low-energy regime and point out the importance of nonuniversal energy scales
inherent to any microscopic model. The spatial dependence of this spectral
function is characterized by oscillatory behavior and an envelope function
which follows a power law both in accordance with the field theoretical
continuum model. Interestingly, for the lattice model we find a phase shift
which is proportional to the two-particle interaction and not accounted for in
the standard bosonization approach to Luttinger liquids with an open boundary.
We briefly comment on the effects of several one-dimensional branches cutting
the Fermi energy and Rashba spin-orbit interaction.
