Real-variable methods on symmetric spaces and Schrodinger operators

对称空间上的实变量方法和薛定谔算子

• 批准号: 0302622
• 负责人: Alexandru Ionescu
• 金额: $ 4.23万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-09-01 至 2004-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0302622&HistoricalAwards=false
• 关键词: Real; variable; methods; symmetric; spaces

ABSTRACT: This project concerns two main topics: harmonic analysis onnoncompact semisimple Lie groups and symmetric spaces, and uniquecontinuation problems and absence of positive eigenvalues of Schrodingeroperators on Euclidean spaces. The author's goal is to developreal-variable methods that could be used to study boundedness propertiesof certain natural operators on noncompact semisimple Lie groups andsymmetric spaces. Some of the problems to be investigated are thefollowing: sharp estimates related to the Kunze-Stein phenomenon,behaviour of the solution of the wave equation on symmetric spaces atlarge time, singular integral operators on symmetric spaces, transferenceprinciples for operators defined by Fourier multipliers on symmetricspaces, and maximal operators and applications in ergodic theory. Theauthor made progress on these problems in certain special cases, mostlyon Lie groups and symmetric spaces of real rank one. The second part ofthe project is aimed at understanding the absence of positive eigenvaluesfor Schrodinger operators with potentials in appropriate Lebesgue spacesand with certain decay properties at infinity.The problem of eliminating the possibility of positive eigenvalues for theSchrodinger operator associated to one or many particles comes frommathematical physics. One expects on physical grounds that such positiveenergy "bound states" cannot exist. This is indeed the case for a largeclass of Schrodinger operators associated to potentials which satisfycertain uniform decay conditions, such as the Coulomb potential.These potentials were studied in the 60's by T. Kato, S. Agmon, J.Weidman, B. Simon, and others. In the last twenty years there has beengrowing interest in extending these classical results to more generalpotentials in various Lebesgue spaces.

摘要：该项目涉及两个主要主题：非紧半单李群和对称空间的调和分析，以及欧几里得空间上薛定谔算子的独特连续问题和正特征值的缺失。作者的目标是开发实变量方法，可用于研究非紧半单李群和对称空间上某些自然算子的有界性质。需要研究的一些问题如下：与昆泽-斯坦现象相关的锐估计、对称空间上波动方程大时间解的行为、对称空间上的奇异积分算子、对称空间上傅里叶乘子定义的算子的传递原理、以及遍历理论中的极大算子和应用。作者在某些特殊情况下对这些问题取得了进展，主要是在李群和实一阶对称空间上。该项目的第二部分旨在了解薛定谔算子在适当的勒贝格空间中具有势和无穷大处具有某些衰变性质的正特征值的缺失。消除与一个或多个粒子相关的薛定谔算子的正特征值的可能性的问题来自数学物理学。从物理角度来看，人们预计这种正能量的“束缚态”不可能存在。对于与满足某些均匀衰变条件的势相关的一大类薛定谔算子来说，确实是这种情况，例如库仑势。T. Kato、S. Agmon、J.Weidman、B. Simon 等人在 60 年代研究了这些势。在过去的二十年里，人们越来越有兴趣将这些经典结果扩展到各种勒贝格空间中更普遍的潜力。