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

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