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Mathematics of Density Functional Theory

Total Cost €


EC-Contrib. €






Project "MDFT" data sheet

The following table provides information about the project.


Organization address
address: RUE MICHEL ANGE 3
city: PARIS
postcode: 75794

contact info
title: n.a.
name: n.a.
surname: n.a.
function: n.a.
email: n.a.
telephone: n.a.
fax: n.a.

 Coordinator Country France [FR]
 Total cost 1˙534˙159 €
 EC max contribution 1˙534˙159 € (100%)
 Programme 1. H2020-EU.1.1. (EXCELLENT SCIENCE - European Research Council (ERC))
 Code Call ERC-2016-COG
 Funding Scheme ERC-COG
 Starting year 2017
 Duration (year-month-day) from 2017-09-01   to  2022-08-31


Take a look of project's partnership.

# participants  country  role  EC contrib. [€] 


 Project objective

Density Functional Theory (DFT) is one of the most famous methods used in quantum physics and chemistry to describe matter at the microscopic scale. The purpose of the proposal is to investigate the mathematical foundations of this theory. The questions which have to be solved involve advanced tools from nonlinear analysis, partial differential equations, spectral theory, optimal transport, and numerical analysis. Several have stayed unsolved for many years. The project is prone to have an impact in many areas of mathematics, as well as in physics and chemistry.

The proposal is divided into three main tasks. The first is focused on some important questions on the foundations of DFT, including excited states, its time-dependent formulation, and the local density approximation based on the uniform electron gas model. DFT can be formulated as an inverse problem, the main question being to find the external potential knowing only the density of particles in the system. We will investigate the invertibility of the potential-to-density map, both in the stationary and time-dependent cases.

The second task deals with the derivation of simple DFT models from the true Schrödinger equation, a problem which has been largely discussed in the literature. We will work on the most challenging open questions, including for instance the relativistic Scott correction, or the proof of Bose-Einstein condensation for an infinite Bose gas in the mean-field limit.

Finally, in the last task we study some particular DFT models, which are particularly challenging from the mathematical point of view. This includes for example a highly nonlinear model for neutrons and protons, infinite crystals with deterministic and random perturbations, and a time-dependent electromagnetic field solving Maxwell's equations coupled to the quantized Dirac quantum vacuum.


year authors and title journal last update
List of publications.
2019 Mathieu Lewin, Elliott H. Lieb, Robert Seiringer
Floating Wigner crystal with no boundary charge fluctuations
published pages: , ISSN: 2469-9950, DOI: 10.1103/PhysRevB.100.035127
Physical Review B 100/3 2020-04-01
2018 Mathieu Lewin, Elliott H. Lieb, Robert Seiringer
Statistical mechanics of the uniform electron gas
published pages: 79-116, ISSN: 2270-518X, DOI: 10.5802/jep.64
Journal de l’École polytechnique — Mathématiques 5 2020-04-01
2018 Arnaud Triay
Derivation of the Dipolar Gross--Pitaevskii Energy
published pages: 33-63, ISSN: 0036-1410, DOI: 10.1137/17m112378x
SIAM Journal on Mathematical Analysis 50/1 2020-04-01
2019 Raphael Ducatez
A forward-backward random process for the spectrum of 1D Anderson operators
published pages: , ISSN: 1083-589X, DOI: 10.1214/19-ECP232
Electronic Communications in Probability 24/0 2020-04-01
2019 Mathieu Lewin, Phan Thành Nam, Nicolas Rougerie
Derivation of renormalized Gibbs measures from equilibrium many-body quantum Bose gases
published pages: 61901, ISSN: 0022-2488, DOI: 10.1063/1.5094331
Journal of Mathematical Physics 60/6 2020-04-01
2020 Mathieu Lewin, Elliott H. Lieb, Robert Seiringer
The local density approximation in density functional theory
published pages: 35-73, ISSN: 2578-5885, DOI: 10.2140/paa.2020.2.35
Pure and Applied Analysis 2/1 2020-04-01
2019 Mathieu Lewin, Peter S. Madsen, Arnaud Triay
Semi-classical limit of large fermionic systems at positive temperature
published pages: 91901, ISSN: 0022-2488, DOI: 10.1063/1.5094397
Journal of Mathematical Physics 60/9 2020-04-01
2019 Louis Garrigue
Hohenberg–Kohn Theorems for Interactions, Spin and Temperature
published pages: 415-437, ISSN: 0022-4715, DOI: 10.1007/s10955-019-02365-6
Journal of Statistical Physics 177/3 2020-04-01
2019 David Gontier, Mathieu Lewin
Spin Symmetry Breaking in the Translation-Invariant Hartree--Fock Electron Gas
published pages: 3388-3423, ISSN: 0036-1410, DOI: 10.1137/19m1243142
SIAM Journal on Mathematical Analysis 51/4 2020-04-01
2020 Ioannis Anapolitanos, Mathieu Lewin
Compactness of Molecular Reaction Paths in Quantum Mechanics
published pages: 505-576, ISSN: 0003-9527, DOI: 10.1007/s00205-019-01475-5
Archive for Rational Mechanics and Analysis 236/2 2020-04-01
2018 Louis Garrigue
Unique Continuation for Many-Body Schrödinger Operators and the Hohenberg-Kohn Theorem
published pages: , ISSN: 1385-0172, DOI: 10.1007/s11040-018-9287-z
Mathematical Physics, Analysis and Geometry 21/3 2019-06-06
2018 Mathieu Lewin
Existence of Hartree–Fock excited states for atoms and molecules
published pages: , ISSN: 0377-9017, DOI: 10.1007/s11005-017-1019-y
Letters in Mathematical Physics 2019-06-06
2018 Mathieu Lewin
Semi-classical limit of the Levy–Lieb functional in Density Functional Theory
published pages: 449-455, ISSN: 1631-073X, DOI: 10.1016/j.crma.2018.03.002
Comptes Rendus Mathematique 356/4 2019-06-06
2019 David Gontier, Christian Hainzl, Mathieu Lewin
Lower bound on the Hartree-Fock energy of the electron gas
published pages: , ISSN: 2469-9926, DOI: 10.1103/physreva.99.052501
Physical Review A 99/5 2019-06-06
2019 Maria Esteban, Mathieu Lewin, Éric Séré
Domains for Dirac–Coulomb min-max levels
published pages: , ISSN: 0213-2230, DOI: 10.4171/rmi/1074
Revista Matemática Iberoamericana 2019-06-06
2018 Lewin , Mathieu; Nam , Phan Thành; Rougerie , Nicolas
The interacting 2D Bose gas and nonlinear Gibbs measures
published pages: , ISSN: , DOI:
Oberwolfach Reports 2019-06-06

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