Double scale analysis of a SchrödingerPoisson system with quantum wells and macroscopic nonlinearities in dimension 2 and 3, Asymptotic Analysis 62

We consider the stationary Schrödinger-Poisson model with a background potential de-scribing a quantum well. The Hamiltonian of this system composes of contributions the background potential well plus a nonlinear repulsive term which extends on di¤erent 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 di¤erent asymptotic behaviours, including spectral renormalization, depending on the dimensions 1, 2 or 3.