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Mathematics of Density Functional Theory

Periodic Reporting for period 2 - MDFT (Mathematics of Density Functional Theory)

Reporting period: 2019-03-01 to 2020-08-31

The project aims at establishing rigorous mathematical results about Density Functional Theory, which is the main method used by chemists and physicists to approximate the many-body Schrödinger equation. The project is divided into three main tasks: foundations of DFT, derivation of DFT approximate theories, and properties of DFT models.
"Several important problems have been solved during the first period. For instance, with Lieb and Seiringer, we have furnished the first rigorous proof of the validity of the Local Density Approximation in DFT and we have solved a longstanding conjecture about the equality of the uniform electron gas and the Jellium model. This was done by introducing a new kind of trial stated describing a ""floating Wigner crystal"". With Gontier and Hainzl, we gave the first quantitative estimates on the energy and critical temperature of an interacting Fermi gas in the Hartree-Fock approximation. This gave some new insight about the form of the phase diagram at high density."
Many other works are ongoing, including for instance on Lieb-Thirring inequalities or on a grand-canonical formulation of optimal transport.
Floating Wigner crystal (Lewin-Lieb-Seiringer, Phys. Rev. B, 100:035127, 2019)
Expected phase diagram of Hartree-Fock Jellium (Gontier-Hainzl-Lewin, Phys. Rev. A, 99:052501, 2019)