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

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.