The interior-boundary Strichartz estimate for the Schr\"odinger equation on the half line revisited

TL;DR

This paper revisits interior-boundary Strichartz estimates for the Schr"odinger equation on the half line, extending their application to low regularity solutions of nonlinear Schr"odinger equations with boundary conditions.

Contribution

It provides a new analysis of interior-boundary Strichartz estimates for the Schr"odinger equation on the half line, including Dirichlet and Neumann cases, and applies these to nonlinear problems.

Findings

Established interior-boundary Strichartz estimates for Schr"odinger on the half line.

Applied estimates to obtain low regularity solutions for nonlinear Schr"odinger equations.

Extended results to coupled systems with boundary conditions.

Abstract

It was shown by the second author in (CPAA, 2019) for the biharmonic Schr\"odinger equation and most recently by Himonas and Mantzavinos (Nonlinearity, 2020) for 2D Schr\"odinger equation that Fokas method based formulas are capable of defining weak solutions of associated nonlinear initial boundary value problems (ibvps) below the Banach algebra threshold. In view of these results, we revisit the theory of interior-boundary Strichartz estimates for the Schr\"odinger equation posed on the right half line, considering both Dirichlet and Neumann cases. Finally, we apply these estimates to obtain low regularity solutions for the nonlinear Schr\"odinger equation (NLS) with Neumann boundary condition and a coupled system of NLS equations defined on the half line with Dirichlet/Neumann boundary conditions.