Scattering for the one dimensional Hartree Fock equation

TL;DR

This paper demonstrates that in 1D Hartree-Fock equations with localized initial data and measure potentials, nonlinear effects prevent long-range scattering, but scattering to linear waves is achieved through space-time resonance analysis.

Contribution

It introduces a novel application of space-time resonances to analyze scattering in 1D Hartree-Fock equations, revealing nonlinear cancellations affecting long-range behavior.

Findings

No long-range scattering due to nonlinear cancellation.

Scattering to linear waves is established.

Space-time resonance framework effectively isolates cancellations.

Abstract

We consider the Hartree-Fock equation in 1D, for a small and localised initial data and a finite measure potential. We show that there is no long range scattering due to a nonlinear cancellation between the direct term and the exchange term for plane waves. We employ the framework of space-time resonances that enables us to single out precisely this cancellation and to obtain scattering to linear waves as a consequence.