Probabilistic pointwise convergence problem of Schrödinger equations on manifolds

Wang, Junfang; Yan, Wei; Yan, Xiangqian

doi:10.1090/proc/15440

Probabilistic pointwise convergence problem of Schrödinger equations on manifolds
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by Junfang Wang, Wei Yan and Xiangqian Yan

Proc. Amer. Math. Soc. 149 (2021), 3367-3378

DOI: https://doi.org/10.1090/proc/15440

Published electronically: May 12, 2021

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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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