Remark on the scattering theory of the nonlinear Schrödinger equation on the cylinders

In this article, we consider the nonlinear Schrödinger equation on the cylinder ${\mathbb{R}}^d\times \mathbb{T}$. In the long range case, we show there is no linear scattering state of the nonlinear Schrödinger equation on ${\mathbb{R}}^d\times \mathbb{T}$. In the short range case, we show the decay and scattering of solutions of the nonlinear Schrödinger equation on ${\mathbb{R}}^d\times \mathbb{T}$ for small data.