Scattering for Defocusing Mass Sub-Critical NLS

In this paper, we consider the L x 2 L_x^2 -scattering of defocusing mass sub-critical nonlinear Schrödinger equations with low weighted initial condition. In prior work, Tsutsumi and Yajima [Bull. Amer. Math. Soc. (N.S.) 11 (1984), no. 1, 186–188, MR741737] established the L 2 L^2 -scattering result for initial data u 0 ∈ H 1 ∩ F H 1 u_0∈ H^1 ∩ {F}H^1 , and this condition was later refined to F H 1 {F}H^1 by Hayashi and Ozawa [Ann. Inst. H. Poincaré Phys. Théor. 48 (1988), no. 1, 17–37, MR 947158 ]. Furthermore, Lee [Int. Math. Res. Not. IMRN 5 (2021), 3571–3596, MR 4227579 ] proved that the inverse wave operator does not exhibit Hölder’s continuity solely within L 2 L^2 , indicating the necessity of introducing a weighted condition for scattering. The determination of how many levels of weighting are required becomes a natural inquiry. Moreover, for large F H s {F}H^s -data with s > 1 s>1 , there only exists the wave operator result, but scattering results are lacking. In the first portion of our result, we establish the L 2 L^2 -scattering within the framework of F H s {F}H^s , where