Scattering in the weighted $L^2$-space for a 2D nonlinear Schr\"odinger equation with inhomogeneous exponential nonlinearity

We investigate the defocusing inhomogeneous nonlinear Schr\"odinger equation $$i \partial_tu + \Delta u = |x|^{-b} \left({\rm e}^{\alpha|u|^2} - 1- \alpha |u|^2 \right) u, \quad u(0)=u_0, \quad x \in \mathbb{R}^2,$$ with $0<b<1$ and $\alpha=2\pi(2-b)$. First we show the decay of global solutions by assuming that the initial data $u_0$ belongs to the weighted space $\Sigma(\mathbb{R}^2)=\{\,u\in H^1(\mathbb{R}^2) \ : \ |x|u\in L^2(\mathbb{R}^2)\,\}$. Then we combine the local theory with the decay estimate to obtain scattering in $\Sigma$ when the Hamiltonian is below the value $\frac{2}{(1+b)(2-b)}$.