Sharp well-posedness of the biharmonic Schrödinger equation in a quarter plane

We obtain an almost sharp local well–posedness result for the biharmonic equation on the quarter plane. In addition, we prove that the nonlinear part of the solution is significantly smoother than the linear part. We use a variant of the restricted norm method of Bourgain adapted to initial–boundary value problems. Our result extends the recent results in Capistrano-Filho et al. (Pacific J Math 309(1):35–70, 2020), Ozsari and Yolcu (Commun Pure Appl Anal 18(6):3285–3316, 2019) and Basakoglu (Part Differ Equ Appl 2(4):37, 2021). It is sharp in the sense that we obtain the well–posedness threshold that was obtained for the full line problem in Seong (J Math Anal Appl 504(1):125342, 2021), with the exception of the endpoint.