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

In this paper we study local and global well-posedness of the following Cauchy problem: $$\bigg \{ \begin{array}{rl} i\partial_t\Psi+\frac{1}{2}\Delta_{x}\Psi = A_0\Psi +\alpha |\Psi|^{\gamma-1}\Psi, & (t,x)\in\R\times\R,\\ (-\Delta_{x})^{\sigma/2}A_0= |\Psi|^2, & \\ \Psi(0,\cdot)=f, & \end{array}$$ with $\sigma\in(0,1)$, $\alpha=\pm 1$, $1<\gamma\leq 5$, in the spaces $L^2(\mathbf{R})$ and $H^1(\mathbf{R})$.