Semiclassical Analysis for Hartree equation

TL;DR

This paper rigorously justifies WKB semiclassical analysis for the Hartree equation in dimensions three and higher, treating the nonlinearity as a semilinear perturbation unlike the quasilinear case in local nonlinear Schrödinger equations.

Contribution

It demonstrates that the Hartree nonlinearity can be handled as a semilinear perturbation in supercritical regimes, simplifying the semiclassical analysis.

Findings

WKB analysis is justified for Hartree in high dimensions.

Nonlinear Hartree term treated as a semilinear perturbation.

Contrast with local nonlinear Schrödinger equations where quasilinear analysis is required.

Abstract

We justify WKB analysis for Hartree equation in space dimension at least three, in a regime which is supercritical as far as semiclassical analysis is concerned. The main technical remark is that the nonlinear Hartree term can be considered as a semilinear perturbation. This is in contrast with the case of the nonlinear Schrodinger equation with a local nonlinearity, where quasilinear analysis is needed to treat the nonlinearity.