Asymptotic analysis of a Schr\"odinger-Poisson system with quantum wells and macroscopic nonlinearities in dimension 1

We consider the stationary one dimensional Schr\"odinger-Poisson system on a bounded interval with a background potential describing a quantum well. Using a partition function which forces the particles to remain in the quantum well, the limit $h\rightarrow0$ in the nonlinear system leads to a uniquely solved nonlinear problem with concentrated particle density. It allows to conclude about the convergence of the solution.