A Kohn-Sham system at zero temperature

A one-dimensional Kohn-Sham system for spin particles is considered which effectively describes semiconductor nanostructures, 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 us to apply Schauder's fixed point theorem. In the 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.