Mathematical Sciences: Regularity of Solutions to Schrodinger Equations

Pi：zhong DMS-9500917这个项目研究了含时薛定谔方程初值柯西问题解的几乎处处性质和局部光滑性。为了研究解的几乎处处性质，利用Stein-Nikishin理论，只需研究相关极大算子的映射性质即可。利用Kolmogrov-Seliverstov-Plessner方法，研究了相应线性化算子的映射性质。解的局部光滑性也是一些线性算子的映射性质。为了研究伴随线性算子的映射性质，我们的方法的主要思想是首先估计核函数，然后应用经典的傅立叶分析理论，如分数次积分的Hardy-Littlewood-Soblev定理、Sobolev嵌入定理、Calderon-Zygmund奇异积分理论和线性解析族的Stein复内插定理。傅立叶变换的Tomas-Stein限制定理与这些映射性质有关。对于某些特殊情况，正交法起着重要的作用。主要的困难是估计核函数。可以应用一些振荡积分估计，如van der Corput型引理。卡莱森、基尼格和鲁伊斯、斯约林、维加、鲁伊斯和维加的想法似乎在这里很重要。薛定谔方程是量子力学中的基本运动方程，是现代物理学的一个重要分支。关于薛定谔方程解的存在性和正则性的结果可以用来解释物理学中的一些现象。1926年1月，薛定谔发明了描述原子离散状态的薛定谔方程。从那时起，薛定谔方程解的存在性和正则性一直是数学和物理学中的重要研究课题。在这个项目中，我们利用现代数学中的一些重要方法来研究薛定谔方程解的正则性。对这些问题的全面理解将在傅立叶分析、偏微分方程组和数学物理等领域有重要的应用。

PI: Zhong DMS-9500917 This project treats the almost everywhere behavior and the local smoothing property of the solutions of the Cauchy problem of the time-dependent Schrodinger equation with the initial value in a Sobolev space.To study the almost everywhere behavior of the solutions, by the Stein-Nikishin theory, it suffices to study the mapping properties of the associated maximal operators. By the Kolmogrov-Seliverstov-Plessner method, it suffices to study the mapping properties of the corresponding linearized operators. The local smoothing properties of the solutions are also the mapping properties of some linear operators. To study the mapping properties of the associated linear operators, the main idea of our approach is to estimate the kernel functions first and then apply classical Fourier analysis theories such as the Hardy-Littlewood-Sobolev theorem for fractional integrals, the Sobolev embedding theorem, the Calderon-Zygmund singular integral theory, and Stein's complex interpolation theorem for an analytic family of linear operators. The Tomas-Stein restriction theorem for the Fourier transform is related to these mapping properties. For certain special case, the orthogonality method plays an important role. The main difficulty is in estimating the kernel functions. Some oscillatory integral estimates, such as van der Corput type lemma, can be applied. The ideas of Carleson, Kenig and Ruiz, Sjolin, Vega, Ruiz and Vega seem to be important here. The Schrodinger equation is the basic equation of motion in Quantum Mechanics, which is an important branch of modern Physics. The results about the existence and the regularity of the solutions of the Schrodinger equation can be used to explain some phenomena in Physics. In January, 1926, Schrodinger invented the Schrodinger equation, which describes the discrete states of an atom. Since then, the existence and the regularity of the solutions of the Schrodinger equation have been i mportant research topics in Mathematics and Physics. In this project, we employ some important techniques in modern Mathematics to study the regularity of the solutions of the Schrodinger equation. The complete understanding of these problems will have some important applications in the areas such as Fourier Analysis, Partial Differential Equations and Mathematical Physics.