Double scale analysis of a Schrödinger–Poisson system with quantum wells and macroscopic nonlinearities in dimension 2 and 3

Abstract

We consider the stationary Schrödinger–Poisson model with a background potential describing a quantum well. The Hamiltonian of this system composes of contributions – the background potential well plus a nonlinear repulsive term – which extends on different length scales with ratio parametrized by the small parameter h. With a partition function which forces the particles to remain in the quantum well, the limit h→0 in the nonlinear system leads to different asymptotic behaviours, including spectral renormalization, depending on the dimensions 1, 2 or 3.