On the modified scattering of $3$-d Hartree type fractional Schrödinger equations with Coulomb potential

Abstract

In this paper we study 3-d Hartree type fractional Schrödin-ger equations: \begin{equation} i\partial_{t}u-|\nabla|^αu = λ\left(|x|^{-γ} *| u|^{2} \right)u,\;\;1 < α< 2,\;\;0 < γ< 3,\;\; λ\in \mathbb R \setminus \{0\}. \end{equation} In \cite{cho} it is known that no scattering occurs in $L^2$ for the long range ($0<\gamma \le 1$). In \cite{c0, chooz2, cho1} the short-range scattering ($1<\gamma <3$) was treated for the scattering in $H^s$. In this paper we consider the critical case ($\gamma =1$) and prove a modified scattering in $L^{\infty }$ on the frequency to the Cauchy problem with small initial data. For this purpose we investigate the global behavior of $x{e}^{it\nabla }u$, $x^2{e}^{it\nabla }u$ and $⟨\xi {⟩}^5\widehat{e^{it\nabla }u}$. Due to the non-smoothness of $\nabla$ near zero frequency the range of $\alpha$ is restricted to $(\frac{17}{10},2)$.

40pages. To appear in Advances in Differential Equations