# Semi-classical limit of the Levy-Lieb functional in Density Functional Theory

## TL;DR

This paper proves that the Levy-Lieb functional in Density Functional Theory converges to a multi-marginal optimal transport problem in the semi-classical limit, extending previous regularization methods to mixed quantum fermionic states.

## Contribution

It extends regularization techniques to mixed fermionic states and establishes the convergence of the Levy-Lieb functional to optimal transport in the semi-classical limit.

## Key findings

- Convergence of Levy-Lieb functional to multi-marginal optimal transport
- Extension of regularization to mixed quantum states
- Applicable to states with or without spin

## Abstract

In a recent work, Bindini and De Pascale have introduced a regularization of $N$-particle symmetric probabilities which preserves their one-particle marginals. In this short note, we extend their construction to mixed quantum fermionic states. This enables us to prove the convergence of the Levy-Lieb functional in Density Functional Theory , to the corresponding multi-marginal optimal transport in the semi-classical limit. Our result holds for mixed states of any particle number $N$, with or without spin.