# Decay and Scattering in energy space for the solution of weakly coupled Schr\"odinger-Choquard and Hartree-Fock equations

## TL;DR

This paper establishes decay estimates and scattering results for solutions to certain non-local Schrödinger equations, including Schrödinger-Choquard and Hartree-Fock systems, using new Morawetz inequalities.

## Contribution

It introduces novel Morawetz inequalities and estimates that enable decay and large-data scattering results for these complex non-local Schrödinger systems.

## Key findings

- Proved decay in Lebesgue norms for non-local Schrödinger equations.
- Established large-data scattering in energy space for Schrödinger-Choquard and Hartree-Fock systems.
- Extended scattering results to any space dimension d ≥ 3.

## Abstract

We prove decay with respect to some Lebesgue norms for a class of Schr\"odinger equations with non-local nonlinearities by showing new Morawetz inequalities and estimates. As a byproduct, we obtain large-data scattering in the energy space for the solutions to the systems of $N$ defocusing Schr\"odinger-Choquard equations with mass-energy intercritical nonlinearities in any space dimension and of defocusing Hartree-Fock equations, for any dimension $d\geq3$.