  We consider a symmetric Anderson impurity model, with a soft-gap
hybridization vanishing at the Fermi level with a power law r > 0. Three facets
of the problem are examined. First the non-interacting limit, which despite its
simplicity contains much physics relevant to the U > 0 case: it exhibits both
strong coupling (SC) states (for r < 1) and local moment (LM) states (for r >
1), with characteristic signatures in both spectral properties and
thermodynamic functions. Second, we establish general conditions upon the
interaction self-energy for the occurence of a SC state for U > 0. This leads
to a pinning theorem, whereby the modified spectral function is pinned at the
Fermi level for any U where a SC state exists; it generalizes to arbitrary r
the familiar pinning condition for the normal r = 0 Anderson model. Finally, we
consider explicitly spectral functions at the simplest level: second order
perturbation theory in U, which we conclude is applicable for r < 1/2 and r > 1
but not for 1/2 < r < 1. Characteristic spectral features observed in numerical
renormalization group calculations are thereby recovered, for both SC and LM
phases; and for the SC state the modified spectral functions are found to
contain a generalized Abrikosov-Suhl resonance exhibiting a characteristic
low-energy Kondo scale with increasing interaction strength.
