Scattering theory in a weighted $L^2$ space for a class of the defocusing inhomogeneous nonlinear Schr\"odinger equation

In this paper, we consider the inhomogeneous nonlinear Schr\"odinger equation (INLS), namely $$i\partial_t u + \Delta u + \mu |x|^{-b} |u|^\alpha u = 0,\quad u(0) = u_0 \in H^1,$$ with $b, \alpha > 0$. We firstly recall a recent result on the local well-posedness for the (INLS) of Guzman [15], and improve this result in the two and three spatial dimensional cases. We next study the decay of global solutions for the defocusing (INLS), i.e. $\mu = −1$ when $0 < \alpha < \alpha^\star$ where $\alpha^\star= \frac{4-2b}{d−2}$ for $d \geq 3$, and $\alpha = \infty$ for $d = 1, 2$ by assuming that the initial data belongs to the weighted $L^2$ space $\Sigma = \{u \in H^1 (\mathbb{R}^d) : |x|u \in L^2 (\mathbb{R}^d)\}$. We finally combine the local theory and the decaying property to show the scattering in $\Sigma$ for the defocusing (INLS) in the case $\alpha_\star < \alpha < \alpha^\star$, where $\alpha_\star = \frac{4−2b}{d}$