We investigate the convergence of the self-consistent field (SCF) iteration used to solve a class of nonlinear eigenvalue problems. We show that for the class of problem considered, the SCF iteration produces a sequence of approximate solutions that contain two convergent subsequences. These subsequences may converge to two different limit points, neither of which is the solution to the nonlinear eigenvalue problem. We identify the condition under which the SCF iteration becomes a contractive fixed point iteration that guarantees its convergence. This condition is characterized by an upper bound placed on a parameter that weighs the contribution from the nonlinear component of the eigenvalue problem. We derive such a bound for the general case as well as for a special case in which the dimension of the problem is 2. 1

Electronic structure and optical properties of metallic nanoshells

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 exchangecorrelation potential uniqueness is shown by monotonicity arguments. Finally, we investigate the behavior of the system if the temperature approaches zero. Subject classification: 34L40, 34L30, 47H05, 81V70 Keywords: Kohn-Sham systems, Schrödinger-Poisson systems, non-linear operators, density operator, zero temperature, Fermi-Dirac distribution