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 which 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, but avoid using any special properties of the potential including the scaling property and screening effects for the localization of the energy.

中文翻译：

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局部密度近似对平滑短程相互作用势的有效性

在全量子理论中，具有给定单体密度的多体量子系统的能量由 Levy-Lieb 泛函描述。它是精确的，但计算起来非常复杂。对于实际计算，引入基于恒定密度的局部能量的局部密度近似是有用的。本文的目的是在 Levy-Lieb 泛函理论和局部密度近似之间建立严格的联系。我们的论证对于在缓慢变化的密度范围内具有一般平滑短程相互作用势的费米子系统是有效的。我们遵循 Lewin、Lieb 和 Seiringer 为库仑势开发的一般方法，但避免使用势的任何特殊属性，包括能量局部化的标度属性和筛选效应。