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On the Convergence of the Selfconsistent Field Iteration for a Class of Nonlinear Eigenvalue Problems
, 2007
"... We investigate the convergence of the selfconsistent 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 subsequ ..."
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We investigate the convergence of the selfconsistent 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
A KohnSham system at zero temperature
, 2008
"... An onedimensional KohnSham 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 Sc ..."
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An onedimensional KohnSham 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 KohnSham potential and obtain W 1,2bounds 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: KohnSham systems, SchrödingerPoisson systems, nonlinear operators, density operator, zero temperature, FermiDirac distribution