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

TL;DR

This paper extends the analysis of scattering theory for long-range Hartree equations with potential |x|^-gamma to a new gamma range, improving the understanding of wave operators without regularity loss.

Contribution

It advances the existing theory by covering a broader gamma range and refining the asymptotic form of solutions for the local Cauchy problem.

Findings

Successfully constructed modified wave operators for 1/3 < gamma < 1/2.

Extended the method from previous work to a new parameter range.

Achieved solutions without loss of regularity in the specified range.

Abstract

We continue the study of the theory of scattering for some long range Hartree equations with potential |x|^-gamma, performed in a previous paper, denoted as I, in the range 1/2 < gamma < 1. Here we extend the results to the range 1/3 < gamma < 1/2. 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 without loss of regularity between the asymptotic state and the solution, as in I, but in contrast to I, we are no longer able to cover the entire subcritical range of regularity of the solutions. The method is an extension of that of I, using a better asymptotic form of the solutions, obtained as the next step of a natural procedure of successive approximations.