Fractional Schr\"odinger-Poisson-Slater system in one dimension

In this paper we study local and global well-posedness of the following Cauchy problem: $$\{\begin{array}{rl}i{\mathrm{\partial }}_t\mathrm{\Psi }+\frac{1}{2}{\mathrm{\Delta }}_x\mathrm{\Psi }=A_0\mathrm{\Psi }+\alpha \mathrm{\mid }\mathrm{\Psi }{\mathrm{\mid }}^{\gamma -1}\mathrm{\Psi },& (t,x)\in \mathbb{R}\times \mathbb{R},\\ (-{\mathrm{\Delta }}_x{)}^{\sigma \mathrm{/}2}{A}_0=\mathrm{\mid }\mathrm{\Psi }{\mathrm{\mid }}^2,& \\ \mathrm{\Psi }(0,\cdot )=f,& \end{array}$$ with $\sigma \in (0,1)$, $\alpha =\pm 1$, $1<\gamma \le 5$, in the spaces $L^2(\mathbf{R})$ and $H^1(\mathbf{R})$.