In this article, we consider the infinite dimensional vector-valued resonant nonlinear Schrödinger system, which arises from the study of the asymptotic behavior of the defocusing nonlinear Schrödinger equation on "wave guide" manifolds like R 2 × T in [7]. We show global 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 [7] closed as crucial ingredients for compactness of the critical elements.The other is the frequency localized interaction Morawetz estimate Theorem 1.3 (Frequency localized interaction Morawetz estimate). Suppose ⃗ u(t, x) is a minimal mass blowup solution to (2.1) on [0, T ] with ∫ T 0 N(t) 3 dt = K. Then (1.4) j∈Z ∇ 1 2 P ≤ 10K ǫ 1 u j (t, x) 2 2 L 2 t,x ([0,T ]×R 2 ) ≲ o(K), where o(K) is a quantity such that o(K) K → 0 as K → ∞.

In this article, we will show the global wellposedness and scattering of the cubic defocusing nonlinear Schrödinger equation on waveguide R 2 ×T in H 1 . We first establish the linear profile decomposition in H 1 (R 2 × T) motivated by the linear profile decomposition of the mass-critical Schrödinger equation in L 2 (R 2 ). Then by using the solution of the infinite dimensional vector-valued resonant nonlinear Schrödinger system to approximate the nonlinear profile, we can prove scattering in H 1 by using the concentrationcompactness/rigidity method.

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