摘要: In this talk, we consider the infinite dimensional vector-valued resonant nonlinear Schr\"odinger system, which arises from the study of the asymptotic behavior of the defocusing  nonlinear Schr\"{o}dinger equation on “wave guide” manifolds  $\mathbb{R}^2\times \mathbb{T}$ in [CHENG,GUO,YANG,ZHAO, Rev. Mat. Iberoam.,2020]. We show globall well-posedness and scattering for this system by long time Strichartz estimates and frequency localized interaction  Morawetz estimates. As a by-product, our results  make the arguments of scattering theory in [CHENG,GUO,YANG,ZHAO, Rev. Mat. Iberoam.,2020] closed as crucial ingredients for compactness of the critical elements. This is a joint work with Prof. Lifeng Zhao.