Simple Models in Mathematical Physics

数学物理中的简单模型

• 批准号: 0701085
• 负责人: Rowan Killip
• 金额: $ 10.45万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0701085&HistoricalAwards=false
• 关键词: Simple; Models; Mathematical; Physics

The project is concerned with three simple models in mathematical physics: random matrices and the linear and non-linear Schrodinger equations. On the first topic, the main goal is to demonstrate the appearance of random matrix statistics in models with comparatively little randomness. Investigations of the (linear) Schrodinger equation will center around the forward/inverse spectral/scattering problems for perturbations of periodic potentials. Global well-posedness of the non-linear Schrodinger equation at critical regularity will be attacked, on the torus and in the focusing case.Consider a drum, not necessarily circluar in shape. When struck, the sound produced is built up of infinitely many components at different frequencies; the exact frequencies are determined by the shape of the drum. (The same is true of other instruments; with differing combinations of frequencies responsible for the characteristic sound of each instrument.) From the list of frequencies, one may determine the area and perimeter of the drum, for example, but not the shape (at least in general). On the basis of numerical investigations, two exciting new conjectures have been made about these characteristic frequencies: when the drum has symmetries, they behave very erratically; while in the absence of symmetries the spacing becomes much more regular. Both statements have precise formulations due to Berry and Tabor in the first instance and Bohigas, Giannoni, and Schmit in the second. A large part of this project is devoted toward making progress on the second conjecture. In particular, in the case of a drum chosen 'at random'.

该项目涉及数学物理中的三个简单模型：随机矩阵和线性和非线性薛定谔方程。 关于第一个主题，主要目标是证明随机矩阵统计量在随机性相对较小的模型中的出现。 （线性）薛定谔方程的研究将集中在周期势扰动的正/逆谱/散射问题。 在临界正则性下，非线性薛定谔方程的整体适定性将在环面上和聚焦的情况下被攻击。考虑一个鼓，不一定是圆形的。 当敲击时，产生的声音是由无限多个不同频率的分量组成的;确切的频率由鼓的形状决定。 (The其他乐器也是如此;每种乐器的特征声音都有不同的频率组合。 从频率列表中，可以确定例如鼓的面积和周长，但不能确定形状（至少一般而言）。 在数值研究的基础上，对这些特征频率进行了两个令人兴奋的新的解释：当鼓具有对称性时，它们的行为非常不规则;而在没有对称性的情况下，间距变得更加规则。 这两种说法都有精确的配方，由于贝里和塔博尔在第一个实例和Bohigas，詹诺尼，和施密特在第二个。 这个项目的很大一部分致力于在第二个猜想上取得进展。 特别是在“随机”选择鼓的情况下。