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, California Institute of TechnologyDMS-0300081----------------------------------------------------Abstract:---------------------------------------------This proposal deals with several problems on the interface between mathematical physics and harmonic analysis.作者打算在确定性和随机潜力的Schroedinger方程式上从事他的工作。与准周期潜力有关的离散施罗丁格运营商仍然存在一些问题，而非独立但更强烈地混合动态的潜力比准周期呈现了许多严重的挑战，并且需要在这方面做更多的挑战。对于时间依赖的方程式，对于分散估计，对于时间依赖和时间独立的电位，问题仍然存在。对于后者，尚不清楚通常的分散估计是否具有比平方逆点更快的电位，至少对于二维尺寸和更大的尺寸。作者最近在一个维度上建立了这一点，但是在二维的分散估计下，在强度多项式衰减的假设下是未知的。但是，他认为，假设有足够快速的衰减潜力的二维情况下的分散估计是一个可访问的问题。线性估计的主要兴趣在于非线性应用。一个例子是由弱相互作用的多索顿解决方案的渐近稳定性证明，这是由Rodnianski，Soffer和作者建立的。它严重依赖电荷传输模型的分散估计。无论是在非线性Schroedinger方程方面（关键的偏置三维方程式的全球解决方案），以及有关非线性结合状态（Solitons）的动态问题的问题，这在该领域仍有很多待在。此外，作者打算在谐波分析中处理问题或在数学物理学以外的问题中进行问题。科学和工程的成功在于其在计算机上的建模和数值研究方面的有效利用数学工具。数学家在开发这些方法并使科学家和工程师提供这些方法方面发挥了重要作用。该建议旨在解决大部分源于数学物理学的数学问题。上述非线性schroedinger方程在各种应用中出现，例如光学。对于这种非线性方程的绑定状态（孤子）代表不瓦解的粒子或梁。一个重要的问题是了解这种对象的稳定性或不稳定。即，它们是否在小扰动下持续存在？显然，在光学介质中，孤子的任何商业应用都需要孤子的稳定性。事实证明，对这些问题的理论理解非常困难，通常需要对数学问题的新见解。该建议旨在解决此类问题。