Smoothing of Transport Plans with Fixed Marginals and Rigorous Semiclassical Limit of the Hohenberg–Kohn Functional

We prove rigorously that the exact N-electron Hohenberg–Kohn density functional converges in the strongly interacting limit to the strictly correlated electrons (SCE) functional, and that the absolute value squared of the associated constrained search wavefunction tends weakly in the sense of probability measures to a minimizer of the multi-marginal optimal transport problem with Coulomb cost associated to the SCE functional. This extends our previous work for N = 2 ( Cotar etal . in Commun Pure Appl Math 66:548–599, 2013 ). The correct limit problem has been derived in the physics literature by Seidl (Phys Rev A 60 4387–4395, 1999 ) and Seidl, Gorigiorgi and Savin (Phys Rev A 75:042511 1-12, 2007 ); in these papers the lack of a rigorous proofwas pointed out.We also give amathematical counterexample to this type of result, by replacing the constraint of given one-body density—an infinite dimensional quadratic expression in the wavefunction—by an infinite-dimensional quadratic expression in the wavefunction and its gradient. Connections with the Lawrentiev phenomenon in the calculus of variations are indicated.