A lower index bilinear estimate for the quadratic Schr\"odinger equation and application for its half line problem

We prove the local well-posedness of the initial boundary value problem for the nonlinear quadratic Schr\"odinger equation under low initial-boundary regularity assumption via the boundary integral operator method introduced by Bona-Sun-Zhang (Trans. Amer. Math. Soc., 354(2):427-490, 2002).