Mathematical Sciences: Regularity of Solutions to Schrodinger Equations

本课题研究Sobolev空间中具有初值的时变薛定谔方程Cauchy问题解的几乎处处性和局部光滑性。为了研究解的几乎处处性，根据Stein-Nikishin理论，研究相关极大算子的映射性质就足够了。利用Kolmogrov-Seliverstov-Plessner方法，研究相应线性化算子的映射性质就足够了。解的局部平滑性质也是一些线性算子的映射性质。为了研究相关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，如分数阶积分的Hardy-Littlewood-Sobolev定理，Sobolev嵌入定理，Calderon-Zygmund奇异积分理论，以及解析线性算子族的Stein复插值定理。傅里叶变换的托马斯-斯坦限制定理与这些映射性质有关。对于某些特殊情况，正交性方法起着重要的作用。主要的困难在于核函数的估计。可以应用一些振荡积分估计，如van der Corput型引理。卡尔森、凯尼格和鲁伊斯、肖林、维加、鲁伊斯和维加的思想在这里似乎很重要。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。薛定谔方程解的存在性和规律性的结果可以用来解释物理学中的一些现象。1926年1月，薛定谔发明了薛定谔方程，它描述了原子的离散状态。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的一个重要研究课题。在这个项目中，我们运用现代数学中的一些重要技术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里叶分析、偏微分方程和数学物理等领域有一些重要的应用。

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.