On the Real Analyticity of the Scattering Operator for the Hartree Equation

TL;DR

This paper proves the real analyticity of the scattering operator for the Hartree equation using novel cut-off and compactness techniques, and simplifies related proofs for the Klein-Gordon equation.

Contribution

It introduces a new method combining cut-offs and compactness to establish analyticity for the Hartree scattering operator, also simplifying Klein-Gordon results.

Findings

Proved real analyticity of the Hartree scattering operator.

Developed a method applicable to Klein-Gordon equation.

Overcame difficulties due to lack of finite speed of propagation.

Abstract

In this paper, we study the real analyticity of the scattering operator for the Hartree equation $ i\partial_tu=-\Delta u+u(V*|u|^2)$. To this end, we exploit interior and exterior cut-off in time and space, and combining with the compactness argument to overcome difficulties which arise from absence of good properties for the nonlinear Klein-Gordon equation, such as the finite speed of propagation and ideal time decay estimate. Additionally, the method in this paper allows us to simplify the proof of analyticity of the scattering operator for the nonlinear Klein-Gordon equation with cubic nonlinearity in Kumlin.