Density of states of a dissipative quantum dot coupled to a quantum wire

We examine the local density of states of an impurity level or a quantum dot coupled to a fractional quantum-Hall edge or to the end of a single one-dimensional Luttinger-liquid lead. Effects of an Ohmic dissipative bath are also taken into account. Using both analytical and numerical techniques we show that, in general, the density of states exhibits power-law frequency dependence near the Fermi energy. In a substantial region of the parameter space it simply reflects the behavior of the tunneling density of states at the end of a Luttinger liquid and is insensitive either to the value of the dot-lead interaction or to the strength of dissipation; otherwise it depends on these couplings too. This behavior should be contrasted with the thermodynamic properties of the level, in particular, its occupancy, which were previously shown to depend on the various interactions in the system only through the corresponding Fermi-edge singularity exponent and thus cannot display any Luttinger-liquid specific power law. Hence, we can construct different models, some with and some without interactions in the wire (but with equal Fermi-edge singularity exponents), which would have very different level densities of states, although they all result in the same level population vs energy curves.