高阶Sobolev范数下具有奇异位势的分数阶Hartree方程的收敛速度
Convergence rate towards the fractional Hartree-equation with singular potentials in higher Sobolev norms
摘要：
本文推广了\cite{AH}和\cite{AHH}的结果。我们想用更自然的假设来证明薛定谔方程向哈特里方程的收敛性。我们首先考虑散焦和聚焦半相对论哈特里方程。我们表明，\cite{P}的工具基本上是足够的导出Hartree方程在这些情况下。接下来，我们将这一结果推广到分数Hartree方程的情况下，可能更多的奇异电位比库仑电位。最后，我们表明，在非相对论性的情况下，可以推导出Hartree方程假设只有$L^2$ -数据的情况下，潜力是超过或作为定期的库仑势。在这种情况下，我们还导出了更多的奇异势的Hartree方程。这项工作的灵感来自于在Sapienza/Rome举行的“MCQM 2018”会议上的演讲。
英文摘要：
This is a work extending the results of \cite{AH} and \cite{AHH}. We want to show convergence of the Schrödinger equation towards the Hartree equation with more natural assumptions. We first consider both the defocusing and the focusing semi-relativistic Hartree equation. We show that the tools of \cite{P} are essentially sufficient for deriving the Hartree equation in those cases. Next, we extend this result to the case of fractional Hartree equations with possibly more singular potentials than the Coulomb potential. Finally, we show that, in the non-relativistic case, one can derive the Hartree equation assuming only $L^2$-data in the case of potentials that are more than or as regular as the Coulomb potential. We also derive the Hartree equation for more singular potentials in this case. This work is inspired by talks given at the conference 'MCQM 2018' at Sapienza/Rome.
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