On the modified scattering of $3$-d Hartree type fractional Schr\"odinger equations with Coulomb potential

TL;DR

This paper establishes modified scattering results for 3D fractional Schrödinger equations with Coulomb potential at the critical decay rate, focusing on small initial data and analyzing the global behavior of solutions in frequency space.

Contribution

It proves modified scattering in $L^ abla$ for the critical case $ ext{γ} = 1$ in 3D fractional Schrödinger equations with Coulomb potential, extending previous results to this challenging regime.

Findings

Proves modified scattering in $L^ abla$ for the critical case $ ext{γ} = 1$.

Analyzes the global behavior of $x e^{it abla} u$, $x^2 e^{it abla} u$, and $raket{ abla}^5 ilde{u}$.

Restricts the fractional order $ ext{α}$ to $(rac{17}{10}, 2)$ due to non-smoothness near zero frequency.

Abstract

In this paper we study 3-d Hartree type fractional Schr\"odin-ger equations: \begin{equation} i\partial_{t}u-|\nabla|^{\alpha}u = \lambda\left(|x|^{-\gamma} *| u|^{2} \right)u,\;\;1 < \alpha < 2,\;\;0 < \gamma < 3,\;\; \lambda \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 $\langle\xi\rangle^5 \widehat{e^{it\nabla} u}$. Due to the non-smoothness of $\nabla$ near zero frequency the range of $\alpha$ is restricted to $(\frac{17}{10}, 2)$.