Scattering for defocusing mass sub-critical NLS

In this paper, we consider the $L_x^2$-scattering of defocusing mass sub-critical nonlinear Schr\"odinger equations with low weighted initial condition. It is known that the scattering holds with $\mathcal{F} H^1$-data, while the continuity of inverse wave operator breaks down with $L^2$-data. Moreover, for large $\mathcal{F} H^s$-data with $s<1$, there only exists the wave operator result, but scattering results are lacking. Our subject is to study the scattering in low weights space. Our results are divided into two parts. Our first result presents a systematic study on the scattering on $\mathcal{F} H^s$ for certain $s<1$, without any restrictions on smallness or radial symmetry. This extends the previous results to spaces with lower weights. Our second result is the almost sure scattering on $L^2$ by introducing a ``narrowed'' Wiener randomization in physical space. For mass subcritical NLS when $d\ge 2$, this result represents the first scattering result without imposing any conditions related to smallness, radial symmetry, or weighted properties on the initial data.