  In this article, we analyze the Dyson equation for the density-density
response function (DDRF) that plays a central role in linear response
time-dependent density functional theory (LR-TDDFT). First, we present a
functional analytic setting that allows for a unified treatment of the Dyson
equation with general adiabatic approximations for discrete (finite and
infinite) and continuum systems. In this setting, we derive a representation
formula for the solution of the Dyson equation in terms of an operator version
of the Casida matrix. While the Casida matrix is well-known in the physics
literature, its general formulation as an (unbounded) operator in the N-body
wavefunction space appears to be new. Moreover, we derive several consequences
of the solution formula obtained here; in particular, we discuss the stability
of the solution and characterize the maximal meromorphic extension of its
Fourier transform. We then show that for adiabatic approximations satisfying a
suitable compactness condition, the maximal domains of meromorphic continuation
of the initial density-density response function and the solution of the Dyson
equation are the same. The results derived here apply to widely used adiabatic
approximations such as (but not limited to) the random phase approximation
(RPA) and the adiabatic local density approximation (ALDA). In particular,
these results show that neither of these approximations can shift the
ionization threshold of the Kohn-Sham system.
