  In nuclear physics, Density Functional Theory (DFT) provides the basis for
state-of-the art studies of ground-state properties of heavy nuclei. However,
the direct relation of the density functional underlying these calculations and
the microscopic nuclear forces is not yet fully understood. We present a
combination of DFT and Renormalization Group (RG) techniques which allows to
study selfbound many-body systems from microscopic interactions. We discuss its
application with the aid of systems of identical fermions interacting via a
long-range attractive and short-range repulsive two-body force in one
dimension. We compute ground-state energies, intrinsic densities, and density
correlation functions of these systems and compare our results to those
obtained from other methods. In particular, we show how energies of excited
states as well as the absolute square of the ground-state wave function can be
extracted from the correlation functions within our approach. The relation
between many-body perturbation theory and our DFT-RG approach is discussed and
illustrated with the aid of the calculation of the second-order energy
correction for a system of $N$ identical fermions interacting via a general
two-body interaction. Moreover, we discuss the control of spuriously emerging
fermion self-interactions in DFT studies within our framework. In general, our
approach may help to guide the development of energy functionals for future
quantitative DFT studies of heavy nuclei from microscopic interactions.
