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A comparative study of the numerical renormalization group and non-crossingapproximation results for the spectral functions of the $U=\infty$ Andersonimpurity model is carried out. The non-crossing approximation is the simplestconserving approximation and has led to useful insights into stronglycorrelated models of magnetic impurities. At low energies and temperatures themethod is known to be inaccurate for dynamical properties due to the appearanceof singularities in the physical Green's functions. The problems in developingalternative reliable theories for dynamical properties have made it difficultto quantify these inaccuracies. As a first step in obtaining a theory which isvalid also in the low energy regime, we identify the origin of the problemswithin the NCA. We show, by comparison with close to exact NRG calculations forthe auxiliary and physical particle spectral functions, that the main source oferror in the NCA is in the lack of vertex corrections in the convolutionformulae for physical Green's functions. We show that the dynamics of theauxiliary particles within NCA is essentially correct for a large parameterregion, including the physically interesting Kondo regime, for all energyscales down to $T_{0}$, the low energy scale of the model, and often well belowthis scale. Despite the satisfactory description of the auxiliary particledynamics, the physical spectral functions are not obtained accurately on scales$\sim T_{0}$. Our results suggest that self--consistent conservingapproximations which include vertex terms may provide a highly accurate way ofdealing with strongly correlated systems at low temperatures.Comment: 33 pages RevTeX, 24 postscript figures, uuencoded fil

Spectral properties of the Anderson impurity model: Comparison of numerical-renormalization-group and noncrossing-approximation results