On Fractional Schrodinger Equations in sobolev spaces

Let $σ\in(0,1)$ with $σ\neq\frac{1}{2}$. We investigate the fractional nonlinear Schrödinger equation in $\mathbb R^d$: $i\partial_tu+(-Δ)^σu+μ|u|^{p-1}u=0,\, u(0)=u_0\in H^s,$ where $(-Δ)^σ$ is the Fourier multiplier of symbol $|ξ|^{2σ}$, and $μ=\pm 1$. This model has been introduced by Laskin in quantum physics \cite{laskin}. We establish local well-posedness and ill-posedness in Sobolev spaces for power-type nonlinearities.

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