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 with complex time in one spatial dimension. Through establishing -maximal estimates for initial datum in , we see that the solution converges to the initial data almost everywhere with when when . 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 -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 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 在一个空间维度上具有复杂的时间。通过建立-中的初始基准的最大估计 ,我们发现解决方案几乎可以在任何地方收敛到初始数据 什么时候 什么时候 。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-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。 与复杂的时间。