On a bilinear restriction estimate for Schrödinger equations on 2D waveguide

In this article, we prove a bilinear estimate for Schrödinger equations on 2d waveguide, $\mathbb{R}\times \mathbb{T}$. We hope it may be of use in the further study of concentration compactness for cubic NLS on $\mathbb{R}\times \mathbb{T}$.