Nonlinear Schr\"odinger equation with Coulomb potential

In this paper, we study the Cauchy problem for the nonlinear Schr\"odinger equations with Coulomb potential $i\partial_tu+\Delta u+\frac{K}{|x|}u=\lambda|u|^{p-1}u$ with $1<p\leq5$ on $\mathbb{R}^3$. We mainly consider the influence of the long range potential $K|x|^{-1}$ on the existence theory and scattering theory for nonlinear Schr\"odinger equation. In particular, we prove the global existence when the Coulomb potential is attractive, i.e. $K>0$ and scattering theory when the Coulomb potential is repulsive i.e. $K\leq0$. The argument is based on the interaction Morawetz-type inequalities and the equivalence of Sobolev norms.