AI 中文总结
本文证明正温度下约化哈特里-福克巨势的半经典极限收敛于托马斯-费米巨势，提出其对偶表示并证明对偶问题的性质，所得结果与零温度情形及费米-狄拉克熵的标准泛函一致，还建立了一致的半经典迹公式。
AI 中文摘要
本文证明，正温度下约化哈特里-福克巨势在半经典极限下收敛于托马斯-费米巨势。特别地，针对一类广泛的费米子熵函数，提出了极限泛函的密度-势对偶表示，对应的对偶问题无对偶间隙且存在极大元。当温度趋于零时，所得巨势收敛于已建立的零温度巨势；对于费米-狄拉克熵，其与标准正温度托马斯-费米自由能泛函一致。该证明还建立了半经典迹公式，该公式在满足共同约束条件的局部预紧势族上是一致的。
英文摘要
It is established that the reduced Hartree-Fock grand potential at positive temperature converges in the semiclassical limit to a Thomas-Fermi grand potential. In particular, a density-potential dual representation of the limiting functional is proposed for a general class of fermionic entropy functions. The corresponding dual problem has no duality gap and admits a maximizer. The resulting grand potential converges to the established zero-temperature grand potential as the temperature tends to zero and, for the Fermi-Dirac entropy, agrees with the standard positive-temperature Thomas-Fermi free-energy functional. The proof also establishes a semiclassical trace formula that is uniform over locally precompact families of potentials subject to a common confinement condition.
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