  We show how spectral functions for quantum impurity models can be calculated
very accurately using a complete set of ``discarded'' numerical renormalization
group eigenstates, recently introduced by Anders and Schiller. The only
approximation is to judiciously exploit energy scale separation. Our derivation
avoids both the overcounting ambiguities and the single-shell approximation for
the equilibrium density matrix prevalent in current methods, ensuring that
relevant sum rules hold rigorously and spectral features at energies below the
temperature can be described accurately.
