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Li, Dong; Xu, Guixiang; Zhang, Xiaoyi
On the Dispersive Estimate for the Dirichlet Schrödinger Propagator and Applications to Energy Critical NLS
We consider the obstacle problem for the Schrödinger evolution in the exterior of the unit ball with Dirichlet boundary condition. Under the radial symmetry we compute explicitly the fundamental solution for the linear Dirichlet Schrödinger propagator $e^{it\Delta_D}$ and give a robust algorithm to prove sharp $L^1 \rightarrow L^{\infty}$ dispersive estimates. We showcase the analysis in dimensions $n=5,7$. As an application, we obtain global well-posedness and scattering for defocusing energy-critical NLS on $\Omega=\mathbb{R}^n\backslash \overline{B(0,1)}$ with Dirichlet boundary condition and radial data in these dimensions.

Keywords:Dirichlet Schrödinger propagator, dispersive estimate, Dirichlet boundary condition, scattering theory, energy critical
Categories:35P25, 35Q55, 47J35

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