  We use the framework set up recently to compute non-perturbatively inelastic
scattering from quantum impurities [G. Zar\'and {\it et al.}, Phys. Rev. Lett.
{\bf 93}, 107204 (2004)] to study the the energy dependence of the single
particle $S$-matrix and the inelastic scattering cross section for a number of
quantum impurity models. We study the case of the spin $S=1/2$ two-channel
Kondo model, the Anderson model, and the usual $S=1/2$ single-channel Kondo
model. We discuss the difference between non-Fermi liquid and Fermi liquid
models and study how a cross-over between the non-Fermi liquid and Fermi liquid
regimes appears in case of channel anisotropy for the $S=1/2$ two-channel Kondo
model. We show that for the most elementary non-Fermi liquid system, the
two-channel Kondo model, half of the scattering remains inelastic even at the
Fermi energy. Details of the derivation of the reduction formulas and a simple
path integral approach to connect the $T$-matrix to local correlation functions
are also presented.
