  The energy-dependent scattering of fermions from a localized orbital at an
energy-dependent rate $\Gamma(\epsilon)\propto |\epsilon|^r$ gives rise to
quantum critical points (QCPs) in the pseudogap single-impurity Anderson model
separating a local moment phase with an unscreened spin moment from a
strong-coupling phase which slightly deviates from the screened phase of
standard Kondo problem. Using the time-dependent numerical renormalization
group (TD-NRG) approach we show that local dynamic properties always
equilibrate towards a steady-state value even for quenches across the QCP but
with systematic deviations from the thermal equilibrium depending on the
distance to the critical coupling. Local non-equilibrium properties are
presented for interaction quenches and hybridization quenches. We augment our
numerical data by an analytical calculation that becomes exact at short times
and find excellent agreement between the numerics and the analytical theory.
For interaction quenches within the screened phase we find a universal function
for the time-dependent local double occupancy. We trace back the discrepancy
between our results and the data obtained by a time-dependent Gutzwiller
variational approach to restrictions of the wave-function ansatz in the
Gutzwiller theory: while the NRG ground states properly account for the
formation of an extended spin moment which decouples from the system in the
unscreened phase, the Gutzwiller ansatz only allows the formation of the spin
moment on the local impurity orbital.
