Simple Models in Mathematical Physics

该项目涉及数学物理中的三个简单模型：随机矩阵以及线性和非线性薛定谔方程。关于第一个主题，主要目标是展示随机矩阵统计在随机性相对较小的模型中的出现。对(线性)薛定谔方程的研究将围绕周期势扰动的正/逆谱/散射问题展开。非线性薛定谔方程在临界正则性下的整体适定性将在环面和聚焦情况下受到攻击。考虑鼓的形状，而不一定是圆形。当敲击时，产生的声音是由不同频率的无限多个分量组成的；确切的频率由鼓的形状决定。(其他乐器也是如此；每种乐器特有的声音都有不同的频率组合。)例如，从频率列表中可以确定鼓的面积和周长，但不能确定鼓的形状(至少在一般情况下)。在数值研究的基础上，对这些特征频率提出了两个令人兴奋的新猜想：当鼓有对称性时，它们的行为非常不规律；而在没有对称性的情况下，间距变得更规则。这两个陈述都有精确的表述，因为第一个是Berry和Tabor，第二个是Bohigas，Giannoni和Schmit。这个项目的很大一部分是致力于在第二个猜想上取得进展。具体地说，在一个鼓被随机选择的情况下。

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'.