  We establish existence and uniqueness of the solution to the Dyson equation
for the density-density response function in time-dependent density functional
theory (TDDFT) in the random phase approximation (RPA). We show that the poles
of the RPA density-density response function are forward-shifted with respect
to those of the non-interacting response function, thereby explaining
mathematically the well known empirical fact that the non-interacting poles
(given by the spectral gaps of the time-independent Kohn-Sham equations)
underestimate the true transition frequencies. Moreover we show that the RPA
poles are solutions to an eigenvalue problem, justifying the approach commonly
used in the physics community to compute these poles.
