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

Article type: Research Article

Authors: Faraj, A. | Mantile, A. | Nier, F.

Affiliations: IMT, UMR – CNRS 5219, Université Paul Sabatier, 31062 Toulouse Cedex 9, France | IRMAR, UMR – CNRS 6625, Université Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France

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.

Keywords: Schrödinger–Poisson systems, asymptotic analysis, multiscale problems, spectral theory

DOI: 10.3233/ASY-2009-0919

Journal: Asymptotic Analysis, vol. 62, no. 3-4, pp. 163-205, 2009