  In a recent letter [Europhys. Lett. 95, 13001 (2011)] the question of whether
the density of a time-dependent quantum system determines its external
potential was reformulated as a fixed point problem. This idea was used to
generalize the existence and uniqueness theorems underlying time-dependent
density functional theory. In this work we extend this proof to allow for more
general norms and provide a numerical implementation of the fixed-point
iteration scheme. We focus on the one-dimensional case as it allows for a more
in-depth analysis using singular Sturm-Liouville theory and at the same time
provides an easy visualization of the numerical applications in space and time.
We give an explicit relation between the boundary conditions on the density and
the convergence properties of the fixed-point procedure via the spectral
properties of the associated Sturm-Liouville operator. We show precisely under
which conditions discrete and continuous spectra arise and give explicit
examples. These conditions are then used to show that in the most physically
relevant cases the fixed point procedure converges. This is further
demonstrated with an example.
