Global attractor for weakly damped Nonlinear Schr\"odinger equations in $L^2(\mathbb{R})$

We prove that the weakly damped nonlinear Schr"odinger flow in $L^2(\mathbb{R})$ provides a dynamical system which possesses a global attractor. The proof relies on the continuity of the Schr"odinger flow for the weak topology in $L^2(\mathbb{R})$.