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 $$i{\mathrm{\partial }}_{t}u+\mathrm{\Delta }u=\mathrm{\mid }x{\mathrm{\mid }}^{-b}({\mathrm{e}}^{\alpha \mathrm{\mid }u{\mathrm{\mid }}^2}-1-\alpha \mathrm{\mid }u{\mathrm{\mid }}^2)u,u(0)=u_0,x\in {\mathbb{R}}^2,$$ 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 $\mathrm{\Sigma }$ when the Hamiltonian is below the value $\frac{2}{(1+b)(2-b)}$.