Global existence and scattering for the inhomogeneous nonlinear Schr\"odinger equation

In this paper we consider the inhomogeneous nonlinear Schr\"odinger equation $i\partial_t u +\Delta u=K(x)|u|^\alpha u,\, u(0)=u_0\in H^s({\mathbb R}^N),\, s=0,\,1,$ $N\geq 1,$ $|K(x)|+|x|^s|\nabla^sK(x)|\lesssim |x|^{-b},$ $0<b<\min(2,N-2s),$ $0<\alpha<{(4-2b)/(N-2s)}$. We obtain novel results of global existence for oscillating initial data and scattering theory in a weighted $L^2$-space for a new range $\alpha_0(b)<\alpha<(4-2b)/N$. The value $\alpha_0(b)$ is the positive root of $N\alpha^2+(N-2+2b)\alpha-4+2b=0,$ which extends the Strauss exponent known for $b=0$. Our results improve the known ones for $K(x)=\mu|x|^{-b}$, $\mu\in \mathbb{C}$ and apply for more general potentials. In particular, we show the impact of the behavior of the potential at the origin and infinity on the allowed range of $\alpha$. Some decay estimates are also established for the defocusing case. To prove the scattering results, we give a new criterion taking into account the potential $K$.