Short-range scattering of Hartree type fractional NLS (Q338443)

In this paper, the author considers a small data scattering problem of the fractional nonlinear Schrödinger equation (fNLS) with Hartree type potential:NEWLINENEWLINENEWLINE\[NEWLINE\begin{cases} i\partial_tu=|\nabla |^\alpha u+(V*|u|^2)u \quad\text{ in }\mathbb{R}^{1+d},\\ u(0)= \varphi \in H^s(\mathbb R^d)\end{cases}NEWLINE\]NEWLINENEWLINENEWLINEwhere \(d\geq 1\), \(1<\alpha<2\), \(s\geq 0\) and \(V\) is a complex-valued measurable function on \(\mathbb R^d\). Here NEWLINE\[NEWLINE|\nabla|^\alpha =|\Delta |^\frac{\alpha}2=\mathcal F^{-1}|\xi|^\alpha\mathcal{F}NEWLINE\]NEWLINE is the fractional derivative of order \( \alpha\) and potential \( V\sim |x|^{-\gamma}\), \(*\) denotes the space convolution. The author shows small data scattering in a weighted space for the short range \(\frac{6-2\alpha}{4-\alpha}<\gamma< 2\). The difficulty arises from the non-locality and non- smoothness of \(|\nabla|^\alpha\). To overcome it they utilize the method of commutator estimate based on Balakrishnan's formula.