Scattering operator for semirelativistic Hartree type equation with a short range potential

Abstract

We prove the existence of the scattering operator in the neighborhood of the origin in $\mathbf{H}^{\omega ,1}\cap \mathbf{H}^{\mu },$ where $\mu > \omega =1+\frac{2}{n}$ for the semirelativistic Hartree type equation \begin{equation*} i\partial _{t}u=\sqrt{M^{2}-\Delta }u+F \big ( u \big ) ,\text{ } \big ( t,x \big ) \in {\mathbf{R}}\times {\mathbf{R}}^{n}, \end{equation*} where $F ( u ) = ( | x | ^{-\gamma }\ast | u | ^{2} ) u,1 < \gamma < 2,n\geq 3$ and $\ast $ denotes the convolution.