  We demonstrate for the first time that a functional-renormalization-group
aided density-functional theory (FRG-DFT) describes well the characteristic
features of the excited states as well as the ground state of an interacting
many-body system with infinite number of particles in a unified manner. The
FRG-DFT is applied to a $(1+1)$-dimensional spinless nuclear matter. For the
excited states, the density--density spectral function is calculated at the
saturation point obtained in the framework of FRG-DFT, and it is found that our
result reproduces a notable feature of the density--density spectral function
of the non-linear Tomonaga-Luttinger liquid: The spectral function has a
singularity at the edge of its support of the lower-energy side. These findings
suggest that the FRG-DFT is a promising first-principle scheme to analyze the
excited states as well as the ground states of quantum many-body systems
starting from the inter-particle interaction.
