一般区域自由费米气体边值问题的定域谱渐近性和关联效应
Localized spectral asymptotics for boundary value problems and correlation effects in the free Fermi gas in general domains
摘要：
在热力学极限下，严格推导了任意维、任意几何形状的自由费米气体基态对关联函数的显式表达式.因此，我们也建立了相应的交换能的局部密度近似的渐近有效性。在恒定密度下，这些公式是通用的，并且不依赖于底层宏观域的几何形状。为了确定在热力学极限的相关效应，我们证明了一个本地外尔定律的光谱渐近的拉普拉斯方程的某些量子观测值，这本身是依赖于一个小参数下非常一般的边界条件。
英文摘要：
We rigorously derive explicit formulae for the pair correlation function of the ground state of the free Fermi gas in the thermodynamic limit for general geometries of the macroscopic regions occupied by the particles and arbitrary dimension. As a consequence we also establish the asymptotic validity of the local density approximation for the corresponding exchange energy. At constant density these formulae are universal and do not depend on the geometry of the underlying macroscopic domain. In order to identify the correlation effects in the thermodynamic limit, we prove a local Weyl law for the spectral asymptotics of the Laplacian for certain quantum observables which are themselves dependent on a small parameter under very general boundary conditions.
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