Scattering Theory in the Energy Space for a Class of Hartree Equations

TL;DR

This paper develops scattering theory in the energy space for Hartree equations with specific potentials, proving asymptotic completeness for radial, nonnegative, and decaying potentials in higher dimensions.

Contribution

It extends scattering theory to a class of Hartree equations with new potential decay and regularity conditions, using Morawetz and Strauss methods.

Findings

Proves asymptotic completeness for certain potentials

Establishes scattering in the energy space for n>2

Handles potentials like |x|^(-γ) with 2<γ<min(4,n)

Abstract

We study the theory of scattering in the energy space for the Hartree equation in space dimension n>2. Using the method of Morawetz and Strauss, we prove in particular asymptotic completeness for radial nonnegative nonincreasing potentials satisfying suitable regularity properties at the origin and suitable decay properties at infinity. The results cover in particular the case of the potential |x|^(- gamma) for 2 < gamma < Min(4,n).