Convergence of the original TDDFT fixed-point iteration scheme

Prove convergence of the original time-dependent density functional theory iteration scheme proposed in the cited fixed-point literature, which corresponds to a nonlinear Schrödinger equation with a loss of two spatial derivatives.

Background

The paper establishes convergence for two modified iteration procedures based on the functional V[Ψ,ρ]. One scheme treats the nonlinearity as a source term, while the other iteratively solves linear Schrödinger equations with potentials generated from the preceding wavefunction; both exploit a reformulation that reduces the effective derivative loss to one derivative.

The original scheme from the cited literature instead corresponds to a nonlinear equation with a loss of two derivatives. The authors explicitly state that their methods do not prove convergence for that original procedure, leaving its convergence unresolved.