Mathematical Sciences: Regularity of Solutions to Schrodinger Equations

数学科学：薛定谔方程解的正则性

• 批准号: 9500917
• 负责人: Jiaping Zhong
• 金额: $ 4.5万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-07-01 至 1998-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9500917&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Regularity; Solutions; Schrodinger

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.

PI：Zhong DMS-9500917 这个项目对待几乎无处不在的行为和局部平滑 非定常微分方程Cauchy问题解的性质 Sobolev空间中的薛定谔方程 解的几乎处处行为，由Stein-Nikishin 理论，它足以研究相关的极大算子的映射性质。通过Kolmogrov-Seliverstov-Plessner方法， 研究相应线性化算子的映射性质。 解的局部光滑性质也是 一些线性算子的性质。 为了研究相关线性算子的映射性质，我们的主要思想是首先估计核函数 然后应用经典的傅里叶分析理论 例如分数积分的Hardy-Littlewood-Sobolev定理， Sobolev嵌入定理，Calderon-Zygmund奇异积分 理论和Stein的解析族复插值定理 线性算子。的Tomas-Stein限制定理 傅立叶变换与这些映射性质有关。对于某些 在特殊情况下，正交方法起着重要的作用。 主要的困难在于核函数的估计。一些振荡 可以应用积分估计，例如货车derCorput型引理。 Carleson，Kenig和Ruiz，Sjolin，Vega，Ruiz和Vega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， 它是现代物理学的一个重要分支。 本文讨论了一类非线性方程组解的存在性和正则性 薛定谔方程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，该方程描述了 原子。从那时起，解的存在性和正则性 薛定谔方程一直是数学界重要研究课题 和物理学。在这个项目中，我们采用了一些重要的技术，在现代 数学中研究薛定谔方程解的正则性 方程对这些问题的全面了解将 在傅里叶分析、偏微分方程和数学物理等领域有着重要的应用。