Sharp well-posedness and ill-posedness results for a quadratic non-linear Schr\"odinger equation

We establish that the quadratic non-linear Schr"odinger equation $$i{u}_t+u_{xx}=u^2$$ where $u:\mathbb{R}\times \mathbb{R}\to \text{C}$, is locally well-posed in $H^s(\mathbb{R})$ when $s\ge -1$ and ill-posed when $s<-1$. Previous work of Kenig, Ponce and Vega had established local well-posedness for $s>-3\mathrm{/}4$. The local well-posedness is achieved by an iteration using a modification of the standard $X^{s,b}$ spaces. The ill-posedness uses an abstract and general argument relying on the high-to-low frequency cascade present in the non-linearity, and a computation of the first non-linear iterate.