Convergence and instability of iterative procedures on the one-dimensional Schrödinger–Poisson problem

Here we study the convergence of numeric solutions for the one-dimensional Schrödinger–Poisson problem for electrons confined into a semiconductor quantum well structure. One kind of algorithm that is largely used is based on a simple iterative procedure that is finished when the solution is achieved when particular parameter (for example, an energy) converges. There is also the possibility of the employ of a mixing parameter to control the variation of a particular parameter of the system, or to fix the number of iterations while a particular parameter of the system is gradually increased (for example, the electron density). We show that the two latter algorithms are capable of solving the problem for a wider class of situations if compared to the former iterative without mixing, without significant loss of precision.