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We consider the Cauchy problem for the fourth order cubic nonlinearSchr\"odinger equation (4NLS). The main goal of this paper is to prove lowregularity well-posedness and mild ill-posedness for (4NLS). We prove threeresults. First, we show that (4NLS) is locally well-posed in$H^s\left(\mathbb{R}\right), s\geq -\frac{1}{2}$ using the Fourier restrictionnorm method. Second, we show that (4NLS) is globally well-posed in$H^s\left(\mathbb{R}\right),s\geq -\frac{1}{2}$. To prove this, we use the$I$-method with the correction term strategy presented inColliander-Keel-Staffilani-Takaoka-Tao [7]. Finally, we prove that (4NLS) ismildly ill-posed in the sense that the flow map fails to be locally uniformlycontinuous in $H^s(\mathbb{R}), s<-\frac{1}{2}$. Therefore, these results showthat $s=-\frac{1}{2}$ is the sharp regularity threshold for which thewell-posedness problem can be dealt with an iteration argument.

Well-Posedness and Ill-Posedness for the Fourth order cubic nonlinear Schr\"odinger equation in negative Sobolev spaces