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

Cotar, C; Friesecke, G; Klüppelberg, C; (2018) Smoothing of Transport Plans with Fixed Marginals and Rigorous Semiclassical Limit of the Hohenberg–Kohn Functional. Archive for Rational Mechanics and Analysis , 228 (3) pp. 891-922. 10.1007/s00205-017-1208-y. Green open access

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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 (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.

Type: Article
Title: Smoothing of Transport Plans with Fixed Marginals and Rigorous Semiclassical Limit of the Hohenberg–Kohn Functional
Open access status: An open access version is available from UCL Discovery
DOI: 10.1007/s00205-017-1208-y
Publisher version: https://doi.org/10.1007/s00205-017-1208-y
Language: English
Additional information: This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions.
Keywords: Density functional theory, Exchange-correlation functional, Optimal transport
UCL classification: UCL
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Statistical Science
URI: https://discovery-pp.ucl.ac.uk/id/eprint/1561733
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