# Smoothing of transport plans with fixed marginals and rigorous semiclassical limit of the Hohenberg-Kohn functional

## TL;DR

This paper rigorously proves the convergence of the N-electron Hohenberg-Kohn functional to the strictly correlated electrons functional in the strong interaction limit, linking density functional theory with optimal transport theory.

## Contribution

It provides the first rigorous proof of the semiclassical limit of the Hohenberg-Kohn functional and establishes the connection with multi-marginal optimal transport with Coulomb cost.

## Key findings

- Convergence of the Hohenberg-Kohn functional to the SCE functional in the strong interaction limit.
- Weak convergence of the wavefunction squared to a minimizer of the optimal transport problem.
- Counterexample showing limitations of certain constraint formulations in the functional.

## Abstract

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 [CFK11]. The correct limit problem has been derived in the physics literature by Seidl [Se99] and Seidl, Gori-Giorgi and Savin [SGS07]; in these papers the lack of a rigorous proof was pointed out. We also give a mathematical 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.