  Within the framework of the full potential projector-augmented wave
methodology, we present a promising low-scaling $GW$ implementation. It allows
for quasiparticle calculations with a scaling that is cubic in the system size
and linear in the number of $k$ points used to sample the Brillouin zone. This
is achieved by calculating the polarizability and self-energy in the real space
and imaginary time domain. The transformation from the imaginary time to the
frequency domain is done by an efficient discrete Fourier transformation with
only a few nonuniform grid points. Fast Fourier transformations are used to go
from real space to reciprocal space and vice versa. The analytic continuation
from the imaginary to the real frequency axis is performed by exploiting
Thiele's reciprocal difference approach. Finally, the method is applied
successfully to predict the quasiparticle energies and spectral functions of
typical semiconductors (Si, GaAs, SiC, and ZnO), insulators (C, BN, MgO, and
LiF), and metals (Cu and SrVO$_3$). The results are compared with conventional
$GW$ calculations. Good agreement is achieved, highlighting the strength of the
present method.
