High-frequency averaging in the semi-classical singular Hartree equation

TL;DR

This paper investigates the semi-classical limit of the nonlinear Schrödinger equation with Hartree nonlinearity, demonstrating the validity of WKB analysis despite singular potentials through high-frequency averaging techniques.

Contribution

It establishes the validity of WKB analysis for the semi-classical Hartree equation with singular potentials, highlighting the role of high-frequency averaging and nonlocal effects.

Findings

WKB analysis remains valid with singular potentials.

No new resonant waves are generated due to nonlinearity.

High-frequency averaging is effective in this nonlocal nonlinear setting.

Abstract

We study the asymptotic behavior of the Schr\"odinger equation in the presence of a nonlinearity of Hartree type in the semi-classical regime. Our scaling corresponds to a weakly nonlinear regime where the nonlinearity affects the leading order amplitude of the solution without altering the rapid oscillations. We show the validity of the WKB-analysis when the potential in the nonlinearity is singular around the origin. No new resonant wave is created in our model, this phenomenon is inhibited due to the nonlinearity. The nonlocal nature of this latter leads us to build our result on a high-frequency averaging effects. In the proof we make use of the Wiener algebra and the space of square-integrable functions.