Modified wave operators without loss of regularity for some long range Hartree equations. I

TL;DR

This paper advances scattering theory for long-range Hartree equations with potential |x|^-gamma, establishing modified wave operators without loss of regularity in the subcritical range, simplifying previous methods.

Contribution

It introduces a new parametrization approach that simplifies the proof of existence of modified wave operators for long-range Hartree equations, avoiding delicate phase estimates.

Findings

Successfully constructs modified wave operators without regularity loss

Extends results to the entire subcritical range 1/2 < gamma < 1

Simplifies proof technique using energy estimates

Abstract

We reconsider the theory of scattering for some long range Hartree equations with potential |x|^-gamma with 1/2 < gamma < 1. More precisely we study the local Cauchy problem with infinite initial time, which is the main step in the construction of the modified wave operators. We solve that problem in the whole subcritical range without loss of regularity between the asymptotic state and the solution, thereby recovering a result of Nakanishi. Our method starts from a different parametrization of the solutions, already used in our previous papers. This reduces the proofs to energy estimates and avoids delicate phase estimates.