  The symmetric Anderson impurity model, with a soft-gap hybridization
vanishing at the Fermi level with power law r > 0, is studied via the numerical
renormalization group (NRG). Detailed comparison is made with predictions
arising from the local moment approach (LMA), a recently developed many-body
theory which is found to provide a remarkably successful description of the
problem. Results for the `normal' (r = 0) impurity model are obtained as a
specific case. Particular emphasis is given both to single-particle excitation
dynamics, and to the transition between the strong coupling (SC) and local
moment (LM) phases of the model. Scaling characteristics and asymptotic
behaviour of the SC/LM phase boundaries are considered. Single-particle spectra
are investigated in some detail, for the SC phase in particular. Here, the
modified spectral functions are found to contain a generalized Kondo resonance
that is ubiquitously pinned at the Fermi level; and which exhibits a
characteristic low-energy Kondo scale that narrows progressively upon approach
to the SC->LM transition, where it vanishes. Universal scaling of the spectra
as the transition is approached thus results. The scaling spectrum
characteristic of the normal Anderson model is recovered as a particular case,
and is captured quantitatively by the LMA. In all cases the r-dependent scaling
spectra are found to possess characteristic low-energy asymptotics, but to be
dominated by generalized Doniach-Sunjic tails, in agreement with LMA
predictions.

Strongly Correlated Electrons