Harmonic Analysis with Applications to Mathematical Physics

调和分析及其在数学物理中的应用

• 批准号: 0617854
• 负责人: Wilhelm Schlag
• 金额: $ 8.55万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-09-01 至 2007-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0617854&HistoricalAwards=false
• 关键词: Harmonic; Analysis; Applications; Mathematical; Physics

PI: Wilhelm Schlag, California Institute of TechnologyDMS-0300081----------------------------------------------------Abstract:---------------------------------------------This proposal deals with several problems on the interface between mathematical physics and harmonic analysis. The author intends to pursue his work on Schroedinger equations with both deterministic and random potentials. Some questions remain on discrete Schroedinger operators on the line with quasi-periodic potentials, whereas the case of potentials given by non-independent but more strongly mixing dynamics than quasi-periodic presents many serious challenges and more needs to be done in this area. For time-dependent equations questions remain concerning dispersive estimates, both for time-dependent and time-independent potentials. For the latter, it is unknown whether or not the usual dispersive estimate holds for potentials that decay faster than an inverse square power, at least for dimensions two and larger. The author recently established this in one dimension, but in two dimensions dispersive estimatesunder the assumption of strong polynomial decay are unknown. He believes, however, that dispersive estimates in the two-dimensional case under the assumption of sufficiently fast decaying potentials is an accessible problem. The main interest in linear estimates lies with nonlinear applications. One example is given by the proof of asymptotic stability of weakly interacting multi-soliton solutions, which was recently established by Rodnianski, Soffer, and the author. It relied heavily on dispersive estimates for charge transfer models. Much remains to be done in this area, both in terms of nonlinear Schroedinger equations in general (global solutions for the critical defocusing three-dimensional equation), as well as questions concerning the dynamics of nonlinear bound states (solitons). In addition, the author intends to work on problems in harmonic analysis or applications thereof to problems outside of mathematical physics.Much of the success of science and engineering lies with its effective use of mathematical tools, both in terms of modeling and numerical studies on computers. Mathematicians play an important role in developing those methods and making them available to scientist and engineers. This proposal aims at addressing mathematical problems that for the most part originate in mathematical physics. The aforementioned nonlinear Schroedinger equations arise in variousapplications, e.g., optics. A bound state (soliton) for such a nonlinear equation represents a particle or beam that travels without disintegrating. An important issue is to understand the stability or instability of such an object. I.e., do they persist under small perturbations or not?Clearly, any commercial application of a soliton in optical media will require stability of the soliton. It turns out that the theoretical understanding of these issues is very difficult, often requiring new insights into mathematical problems. This proposal aims at addressing such problems.

PI：Wilhelm Schlag，加州理工学院DMS-0300081---------------------------------------------------- - - - - - - -抽象的： - - - - - - - - - - - - - - - - - - ----------该提案涉及数学物理和调和分析之间的接口上的几个问题。作者打算继续研究具有确定性和随机势的薛定谔方程。准周期势线上的离散薛定谔算子仍然存在一些问题，而由非独立但比准周期更强烈的混合动力学给出的势的情况提出了许多严峻的挑战，并且在该领域需要做更多的工作。对于时间相关方程，关于时间相关和时间无关势的色散估计的问题仍然存在。对于后者，目前尚不清楚通常的色散估计是否适用于比平方反幂次方衰减更快的势能，至少对于二维和更大的维度来说是这样。作者最近在一维中建立了这一点，但在强多项式衰减假设下的二维色散估计是未知的。然而，他认为，在假设势能衰减足够快的情况下，二维情况下的色散估计是一个容易解决的问题。线性估计的主要兴趣在于非线性应用。弱相互作用多孤子解的渐近稳定性证明给出了一个例子，这是由 Rodnianski、Soffer 和作者最近建立的。它严重依赖于电荷转移模型的分散估计。在这个领域还有很多工作要做，无论是一般的非线性薛定谔方程（临界散焦三维方程的全局解），还是有关非线性束缚态（孤子）动力学的问题。此外，作者打算研究调和分析中的问题或其在数学物理之外的问题中的应用。科学和工程的成功很大程度上取决于数学工具的有效使用，无论是在计算机建模还是数值研究方面。数学家在开发这些方法并将其提供给科学家和工程师方面发挥着重要作用。该提案旨在解决大部分源自数学物理学的数学问题。上述非线性薛定谔方程出现在各种应用中，例如光学。这种非线性方程的束缚态（孤子）表示在不分解的情况下传播的粒子或光束。一个重要的问题是了解此类物体的稳定性或不稳定性。即，它们在小扰动下是否持续存在？显然，孤子在光学介质中的任何商业应用都需要孤子的稳定性。事实证明，从理论上理解这些问题非常困难，往往需要对数学问题有新的见解。本提案旨在解决此类问题。