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

Nonlinear Schrodinger 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 semiclassical limit of the three-dimensional (3D) Schrodinger--Poisson system with an additional harmonic potential, we study some semiclassical limits of the Hartree equation with harmonic potential in space dimension $n \geq 2$. 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 Schrodinger--Poisson system with harmonic potential, we can give only a formal computation since the need for modified scattering operators for this long-range scattering case goes beyo...

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