With regard to Jerome's paper: Can this be extended to unbounded regions? Also, why are homogeneous Dirichlet boundary values imposed? Are these the most natural? Can others be used, in particular, nonzero? Additionally, he mentions at the end a special case: dimension 1, where one might need the Hilbert transform to deal with the Hartree potential.

With regard to the Sprengel papers (arXiv:1701.02679, also seconadrily arXiv:1701.02679): It seems one can extend this more or less straightforwardly to adiabatic GGA, yet no non-adiabatic (history) terms in the effective potentials, which are also considered by Jerome, seem to be possible.

The strong correlation limit of (time-independent) DFT, i.e. interactions dominate the kinetic energy, can be reformulated as a classical Optimal Transport (Monge–Kantorovich) problem with the Coulomb repulsion $1/|x-y|$ as a cost function, see arXiv:1205.4514.

In a setting without interactions (like a KS system) the whole time-dependent Schrödinger equation is actually a classical Newtonian law on a symplectic manifold with Wasserstein metric (ths relating to a optimal transport problem with the usual $|x-y|^2$ cost function) - this follows from a little known work of Max von Renesse that builds on a fluid dynamics reformulation of optimal transport due to Benamou-Brenier.

Can one reformulate the Optimal Control of Sprengel as Optimal Transport including time, so the target is a function in spacetime and one moves towards it in some auxiliary time variable?