We present some progress in the direction of determining the semiclassical limit of the Hoenberg-Kohn universal functional in Density Functional Theory for Coulomb systems. In particular we give a proof of the fact that for Bosonic systems with an arbitrary number of particles the limit is the multimarginal optimal transport problem with Coulomb cost and that the same holds for Fermionic systems with 2 or 3 particles. Comparisons with previous results are reported . The approach is based on some techniques from the optimal transportation theory.
We consider a multimarginal transport problem with repulsive cost, where the marginals are all equal to a fixed probability ρ ∈ P(R d ). We prove that, if the concentration of ρ is less than 1/N , then the problem has a solution of finite cost. The result is sharp, in the sense that there exists ρ with concentration 1/N for which the cost is infinite.
In this paper we study the three-marginal optimal mass transportation problem for the Coulomb cost on the plane R 2 . The key question is the optimality of the so-called Seidl map, first disproved by Colombo and Stra. We generalize the partial positive result obtained by Colombo and Stra and give a necessary and sufficient condition for the radial Coulomb cost to coincide with a much simpler cost that corresponds to the situation where all three particles are aligned. Moreover, we produce an infinite family of regular counterexamples to the optimality of Seidl-type maps.
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