The validity of the local density approximation for smooth short range interaction potentials

Abstract

In the full quantum theory, the energy of a many-body quantum system with a given one-body density is described by the Levy–Lieb functional. It is exact but very complicated to compute. For practical computations, it is useful to introduce the local density approximation that is based on the local energy of constant densities. The aim of this paper is to make a rigorous connection between the Levy–Lieb functional theory and the local density approximation. Our justification is valid for fermionic systems with a general class of smooth short range interaction potentials, in the regime of slowly varying densities. We follow a general approach developed by Lewin, Lieb, and Seiringer for Coulomb potential [M. Lewin et al., Pure Appl. Anal. 2(1), 35–73 (2020)] but avoid using any special properties of the potential including the scaling property and screening effects for the localization of the energy.