Small data scattering for semi-relativistic equations with Hartree type nonlinearity

We prove that the initial value problem for the equation $- i\partial_t u + \sqrt{m^2-Δ} \, u= (\frac{e^{-μ_0 |x|}}{|x|} \ast |u|^2)u \ \text{in} \ \mathbb R^{1+3}, \quad m\ge 0, \ μ_0 >0$ is globally well-posed and the solution scatters to free waves asymptotically as $t \to \pm \infty$ if we start with initial data which is small in $H^s(\mathbb R^{3})$ for $s>\frac12$, and if $m>0$. Moreover, if the initial data is radially symmetric we can improve the above result to $m\ge 0$ and $s>0$, which is almost optimal, in the sense that $L^2(\mathbb R^{3})$ is the critical space for the equation. The main ingredients in the proof are certain endpoint Strichartz estimates, $L^2(\mathbb R^{1+3})$ bilinear estimates for free waves and the application of the $U^p$ and $V^p$ function spaces.

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