A comparative study of the numerical renormalization group and non-crossing approximation results for the spectral functions of the $U=infty$ Anderson impurity model is carried out. The non-crossing approximation is the simplest conserving approximation and has led to useful insights into strongly correlated models of magnetic impurities. At low energies and temperatures the method is known to be inaccurate for dynamical properties due to the appearance of singularities in the physical Greens functions. The problems in developing alternative reliable theories for dynamical properties have made it difficult to quantify these inaccuracies. As a first step in obtaining a theory which is valid also in the low energy regime, we identify the origin of the problems within the NCA. We show, by comparison with close to exact NRG calculations for the auxiliary and physical particle spectral functions, that the main source of error in the NCA is in the lack of vertex corrections in the convolution formulae for physical Greens functions. We show that the dynamics of the auxiliary particles within NCA is essentially correct for a large parameter region, including the physically interesting Kondo regime, for all energy scales down to $T_{0}$, the low energy scale of the model, and often well below this scale. Despite the satisfactory description of the auxiliary particle dynamics, the physical spectral functions are not obtained accurately on scales $sim T_{0}$. Our results suggest that self--consistent conserving approximations which include vertex terms may provide a highly accurate way of dealing with strongly correlated systems at low temperatures.
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