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

Abstract

Abstract. 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.

Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice – III. From ab initio models to WKB for Schrödinger–Poisson

“…This is somewhat a homogenization process for the Poisson equation arising in this 1d context. In a 3-D case, a stationary phase argument on the Hartree term would (formally) give a similar result with m = 1 (see also [19,40]). …”

“…(18) and (19) mean in particular that an initial datum in the nth band E n ðjÞ; z n j always leads to an approximate solution in the same band; hence the Poisson nonlinearity induces a behaviour not so different compared to the linear cases in [26]. This is one of the reasons why it is possible to extend K-branch solutions to cover the present situation.…”

“…[14,16,24,25,46], have been used up to now exclusively in the context of linear dispersive equations. Hence, we must show now how to adapt this tool in order to tackle the weakly nonlinear problem (19).…”

“…One sees here one big advantage in using WKB methods in the present context when compared to Wigner techniques. Indeed, WKB construction delivers a rather simple expression (18) for a wave function belonging to the nth energy band and approximately satisfying (1) for t P 0 and e positive but small enough through the solving of (19). In sharp constrast, Wigner techniques furnish only information on the ideal e = 0 case discarding fine-scale modulations z n j ðÁ=eÞ; it would therefore be more difficult to study numerically the weak consistency of physical observables relying on this approach, left apart the more singular moment systems admitting measure-valued solutions, see [27,33,46,49].…”

Cascade of Phase Shifts and Creation of Nonlinear Focal Points for Supercritical Semiclassical Hartree Equation

“…One is concerned with the nonlinear effect on the behavior far from the focal point, and the other with that near the focal point. The situation is similar in the case of Hartree equation (see, [8,10,27]). The two critical indices are α = 1 (far from focal point) and α = γ (near the focal point).…”