Abstract:In this paper, we investigate the probabilistic pointwise convergence problem of Schrödinger equation on the manifolds. We prove probabilistic pointwise convergence of the solutions to Schrödinger equations with the initial data in $L^{2}(\mathrm {\mathbf {T}}^{n})$, where $\mathrm {\mathbf {T}}=[0,2\pi )$, which require much less regularity for the initial data than the rough data case. We also prove probabilistic pointwise convergence of the solutions to Schrödinger equation with Dirichlet boundary condition for a large set of random initial data in $\cap _{s<\frac {1}{2}}H^{s}(\Theta )$, where $\Theta$ is three dimensional unit ball, which require much less regularity for the initial data than the rough data case.

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中文翻译：

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流形上薛定谔方程的概率逐点收敛问题

摘要：在本文中，我们研究了薛定谔方程在流形上的概率逐点收敛问题。我们用 $L^{2}(\mathrm {\mathbf {T}}^{n})$ 中的初始数据证明了薛定谔方程解的概率逐点收敛，其中 $\mathrm {\mathbf {T}} =[0,2\pi )$，与粗略数据情况相比，初始数据需要的规律性要少得多。我们还证明了在 $\cap _{s<\frac {1}{2}}H^{s}(\Theta ) 中的大量随机初始数据的 Schrödinger 方程解的概率逐点收敛与 Dirichlet 边界条件$，其中 $\Theta$ 是三维单位球，与粗略数据情况相比，初始数据需要的规律性要少得多。

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