Global existence and scattering for the inhomogeneous nonlinear Schrödinger equation

In this paper, we consider the inhomogeneous nonlinear Schrödinger equation \(i\partial _t u +\Delta u =K(x)|u|^\alpha u,\; u(0)=u_0\in H^1({\mathbb {R}}^N),\; N\ge 3,\; |K(x)|+|x||\nabla K(x)|\lesssim |x|^{-b},\; 0<b< \min (2, N-2),\; 0<\alpha <{(4-2b)/(N-2)}\). 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 \(K(x)=\mu |x|^{-b}\), \(\mu \in {\mathbb {C}}\). For general potentials, we highlight the impact of the behavior at the origin and infinity on the allowed range of \(\alpha \). In the defocusing case, we prove decay estimates provided that the potential satisfies some rigidity-type condition which leads to a scattering result. We give also a new scattering criterion taking into account the potential K.