(Semi)Classical Limit of the Hartree Equation with Harmonic Potential

Nonlinear Schrödinger Equations (NLS) of the Hartree type occur in the modeling of quantum semiconductor devices. Their ”semiclassical” limit of vanishing (scaled) Planck constant is both a mathematical challenge and practically relevant when coupling quantum models to classical models. With the aim of describing the semi-classical limit of the 3D Schrödinger–Poisson system with an additional harmonic potential, we study some semi-classical limits of the Hartree equation with harmonic potential in space dimension n ≥ 2. The harmonic potential is confining, and causes focusing periodically in time. We prove asymptotics in several cases, showing different possible nonlinear phenomena according to the interplay of the size of the initial data and the power of the Hartree potential. In the case of the 3D Schrödinger–Poisson system with harmonic potential, we can only give a formal computation since the need of modified scattering operators for this long range scattering case goes beyond current theory. We also deal with the case of an additional ”local” nonlinearity given by a power of the local density a model that is relevant when incorporating the Pauli principle in the simplest model given by the ”Schrödinger-Poisson-Xα equation”. Further we discuss the connection of our WKB based analysis to the Wigner function approach to semiclassical limits.