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

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• Response potential in the strong-interaction limit of DFT: Analysis and comparison with the coupling-constant average
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• Universal Functionals in Density Functional Theory
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• Functional Derivative of the Zero Point Energy Functional from the Strong Interaction Limit of Density Functional Theory
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• From wave-functions to single electron densities
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• Investigation of the exchange-correlation potential of functionals based on the adiabatic connection interpolation
• Marginals with finite repulsive cost
• Uniform approximation of continuous couplings