Mage's Rhizome

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====== Notes Eric + Markus ======

==== Kohn-Sham equations ====


===Jerome's Paper===
  * With regard to [[http://sites.math.northwestern.edu/~jwj/preprints/JMAA/JMAA.pdf|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.

  * There is also the previous result of Jerome here [[https://arxiv.org/pdf/1309.3587.pdf|Jerome's previous paper]] which is more numerical in nature. Here bounded domains with homogeneous B.C. are also studied. 

  * 1D Hartree potential for the electron charge density $\rho = |\psi|^2$: $$V_H(x,t) = 1/|x| \ast \rho (x,t) = \int_a^b \dfrac{\rho(y,t)}{|x-y|} dy$$ and Hilbert transform  for $f \in L^2 (\mathbb{R})$ is $$H f (y) = \lim_{\epsilon \to 0} \dfrac{1}{\pi} \int_{|x| > \epsilon} \dfrac{f(y-x)}{x} dx$$ Recall that $H: L^2 (\mathbb{R}) \to L^2 (\mathbb{R})$ is an isometry and for $1<p < \infty$, $H: W^{s,p}(\mathbb{R}) \to W^{s,p}(\mathbb{R})$ is an isomorphism. In the 1D case Jerome suggests considering the softened potential given (e.g. for Helium atom) by $$V(x_1, x_2) = \dfrac{1}{\sqrt{(x_1-x_2)^2+1}}$$ subject to a softened external potential $$V_{\text{ext}}(x) = \dfrac{-2}{\sqrt{x^2+1}}$$ Jerome suggests Hilbert transform is needed for this, but maybe if another type of softened potential is considered, e.g. Eric's $V(x) = \dfrac{e^{-C/|x|}}{|x|}$ no Hilbert transform is needed?
 
  * Ionic potentials not included in Jerome's analysis, can we include them? What general form do they take? 




===Sprengel Papers===

  * With regard to the Sprengel papers ([[https://arxiv.org/abs/1701.02679|arXiv:1701.02679]], also seconadrily [[https://arxiv.org/abs/1701.02679|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.

==== Optimal transport ====

  * 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 [[https://arxiv.org/abs/1205.4514|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 [[https://www.mathematik.uni-muenchen.de/~renesse/Docs/SCHROED.pdf|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?

==== Improved Runge-Gross theorem ====

  * Markus' main current idea is to use the [[http://iopscience.iop.org/article/10.1209/0295-5075/95/13001|fixed-point Runge-Gross proof]] and apply it to TD current DFT. Although the current state of the fixed-point proof looks convincing, this is a little bit deceiving, since the norms in the final stage of the assumed contraction mapping do not really match. It is expected that they match in the CDFT setting where the current enters as an additional reduced quantity, further the involved PDE is only of order 1 and allows a method of characteristics approach. The original fixed-point proof was conceived to circumvent problems with non-analyticity in time, but some form of time regularity seems to be necessary (and will be introduced automatically through the use of the current).