On Strichartz estimate for many body Schr\"odinger equation in the waveguide setting

In this short paper, we prove Strichartz estimates for N-body Schr\"odinger equations in the waveguide manifold setting (i.e. on semiperiodic spaces $\mathbb{R}^m\times \mathbb{T}^n$ where $m\geq 3$), provided that interaction potentials are small enough (depending on the number of the particles and the universal constants, not on the initial data). The proof combines both the ideas of Tzvetkov-Visciglia \cite{TV1} and Hong \cite{hong2017strichartz}. As an immediate application, the scattering asymptotics for this model is also obtained. This result extends Hong \cite{hong2017strichartz} to the waveguide case.