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On the dimension of divergence sets of Schrödinger equation with complex time
Nonlinear Analysis ( IF 1.4 ) Pub Date : 2021-02-27 , DOI: 10.1016/j.na.2021.112312
Jiye Yuan , Tengfei Zhao , Jiqiang Zheng

This article studies the pointwise convergence for the fractional Schrödinger operator Pa,γt with complex time in one spatial dimension. Through establishing L2-maximal estimates for initial datum in Hs(R), we see that the solution converges to the initial data almost everywhere with s>14a(11γ)+ when 012(11γ)+ when a=1. By constructing counterexamples, we show that this result is almost sharp up to the endpoint. These results extend the results of P. Sjölin, F. Soria and A. Bailey. Second, we study the Hausdorff dimension of the set of the divergent points, by showing some L1-maximal estimates with respect to general Borel measure. Our results reflect the interaction between dispersion effect and dissipation effect, arising from the fractional Schrödinger type operator Pa,γt with the complex time.



中文翻译:

具有复杂时间的薛定ding方程的发散集的维数

本文研究分数Schrödinger算子的逐点收敛 P一种γŤ在一个空间维度上具有复杂的时间。通过建立大号2个-中的初始基准的最大估计 Hs[R,我们发现解决方案几乎可以在任何地方收敛到初始数据 s>1个4一种1个-1个γ+ 什么时候 01个2个1个-1个γ+ 什么时候 一种=1个。通过构造反例,我们表明,此结果几乎到端点为止都是清晰的。这些结果扩展了P.Sjölin,F。Soria和A. Bailey的结果。其次,通过显示一些散点集,我们研究其Hausdorff维数大号1个-关于一般Borel测度的最大估计。我们的结果反映了分数Schrödinger型算子引起的色散效应和耗散效应之间的相互作用。P一种γŤ 与复杂的时间。

更新日期:2021-02-28
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