Fractional Schrödinger-Poisson-Slater system in one dimension

In this paper we study local and global well-posedness of the following Cauchy problem: $\bigg \{ \begin{array}{rl} i\partial_tΨ+\frac{1}{2}Δ_{x}Ψ= A_0Ψ+α|Ψ|^{γ-1}Ψ, & (t,x)\in\R\times\R,\\ (-Δ_{x})^{σ/2}A_0= |Ψ|^2, & \\ Ψ(0,\cdot)=f, & \end{array}$ with $σ\in(0,1)$, $α=\pm 1$, $1<γ\leq 5$, in the spaces $L^2(\mathbf{R})$ and $H^1(\mathbf{R})$.

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