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

We consider the stationary Schrodinger-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.

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