零温下的Kohn-Sham系统
A Kohn-Sham system at zero temperature
摘要：
本文考虑了一个自旋粒子的一维Kohn-Sham系统，它有效地描述了半导体纳米结构，并在零温度下进行了研究。我们证明了解决方案的存在性，并得出先验估计。为此，我们找到了具有有效Kohn-Sham势的薛定谔算子的本征值的估计，并得到了伴随粒子密度算子的$W^{1,2}$ -界。之后，紧性和连续性的结果允许应用Schauder的不动点定理。在交换相关势为零的情况下，单调性参数显示了唯一性。最后，我们研究了当温度接近零时系统的行为。
英文摘要：
An one-dimensional Kohn-Sham system for spin particles is considered which effectively describes semiconductor {nano}structures and which is investigated at zero temperature. We prove the existence of solutions and derive a priori estimates. For this purpose we find estimates for eigenvalues of the Schrödinger operator with effective Kohn-Sham potential and obtain $W^{1,2}$-bounds of the associated particle density operator. Afterwards, compactness and continuity results allow to apply Schauder's fixed point theorem. In case of vanishing exchange-correlation potential uniqueness is shown by monotonicity arguments. Finally, we investigate the behavior of the system if the temperature approaches zero.
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