Calculations of the particle dynamics in accelerators beyond the usual perturbation techniques

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 Green's 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.