WIAS Preprint No. 1293, (2008)

A Kohn--Sham system at zero temperature


  • Cornean, Horia
  • Hoke, Kurt
  • Neidhardt, Hagen
  • Racec, Paul N.
  • Rehberg, Joachim

2010 Mathematics Subject Classification

  • 34L40 34L30 47H05 81V70


  • Kohn-Sham systems, Schrödinger-Poisson systems, non-linear operators, density operator, zero temperature, Fermi-Dirac distribution




An 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 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.

Appeared in

  • J. Phys. A, 41 (2008) pp. 385304/1--385304/21.

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