Scaling limits in the 3-D Schrödinger-Poisson system Original Research Article

The equivalence between the Schrödinger formalism for pure quantum states and the Wigner representation of quantum mechanics, and the importance of the latter in semiconductor modelling motivates the analysis of the large-time behaviour for the Schrödinger-Poisson problem arising in this context. Using an appropriate rescaling group and some self-similarity properties of the rescaled solutions, it is proved that the kinetic energy effects in the rescaled system vanish for large times. Then, when the time goes to infinity, the limit potential of the original system can be identified as a Coulomb potential.