Spectral and Asymptotoc Problems of Mathematical Physics

哈雷尔9622730这个项目是关于在量子物理和反应扩散中出现的微分算符的谱理论中的几个问题的数学分析。希望这些结果将对物理学家真正有用，同时仍停留在严格和原创的泛函分析或微分方程式理论的领域。许多特殊的问题都与共振和亚稳态有关，或者与几何、算符代数和光谱之间的联系有关。提案中的第一个具体主题涉及势能与曲率有关的薛定谔算子的本征值。这听起来很抽象，但它实际上源于对双稳金属合金相区形状的分析。第二个主题着眼于Harrell和Stubbe最近发现的某些纯代数恒等式，这些恒等式被证明非常适用于配分函数(=热核的踪迹)以及其他谱表达式。第三个主题与半经典分析有关，哈雷尔在这个领域已经工作了几年，但新的特征包括磁场和依赖时间的相互作用。最后一个主题涉及到本征函数的局部性质，例如它们的节点，以及涉及它们的一些反问题。需要使用的数学机制包括微扰理论、渐近性和变分分析。这个项目涉及对物理学中的几个问题的数学分析，特别是原子、分子、半导体和金属合金的物理学。许多特殊的问题都与共振和亚稳态(缓慢变化)现象有关，还有一些问题与几何学在这些物理领域中的作用有关。这项研究的目的是为工作中的物理学家提供一些新的和有用的东西，同时阐明物理理论所依赖的数学结构。L的建议中的第一个具体主题是估计像量子力学中的方程的“简正模”，但在势能涉及曲率的情况下。这听起来像是一个抽象的数学问题，但它实际上源于对具有两个不同稳定相的金属区域的分析，当人们试图找到具有一个或另一个相的区域的形状时。第二个主题着眼于Harrell和Stubbe最近发现的某些纯代数恒等式，事实证明这些恒等式适用于统计力学的中心函数之一。第三个问题与量子力学理论和经典物理理论之间的差异有关，经典物理理论在更大范围内适用于我们的世界。哈雷尔在这个领域已经工作了几年；新的创新包括融入磁性。最后一个问题与量子力学波函数的一些鲜为人知的局域性质有关，以及测量它们如何确定存在什么样的力。所使用的方法既来自纯数学，也来自应用数学，以及物理学。

Abstract Harrell 9622730 This project is concerned with the mathematical analysis of several problems in spectral theory of differential operators arising in quantum physics and reaction-diffusion. It is hoped that the results will be of real use to physicists, while remaining in the realm of rigorous and original functional analysis or differential-equations theory. Many of the particular issues have to do with resonances and metastability, or with connections among geometry, operator algebra, and spectra. The first specific topic in the proposal concerns the eigenvalues of Schroedinger operators where the potential energy involves curvature. This sounds abstract, but it actually originates in the analysis of the shapes of phase regions of bistable metallic alloys. The second topic looks at certain purely algebraic identities recently discovered by Harrell and Stubbe, which turn out to apply, sharply, to the partition function (= trace of the heat kernel), among other spectral expressions. The third topic has to do with semiclassical analysis, an area in which Harrell has worked for several years, but new features include magnetic fields and time-dependent interactions. The final topic has to do with local properties of eigenfunctions, such as their nodes, and some inverse problems involving them. Mathematical machinery to be called upon ncludes perturbation theory, asymptotics, and variational analysis. This project is concerned with the mathematical analysis of several problems in physics, especially the physics of atoms, molecules, semiconductors, and metallic alloys. Many of the particular problems have to do with resonances and with metastable (slowly changing) phenomena, and others have to do with the role of geometry in these areas of physics. The goal of the research is to produce something new and useful to working physicists and at the same time to elucidate the mathematical structure which the physical theory relies on. The first specific topic in the proposa l is to make estimates of the "normal modes" of an equation like those in quantum mechanics, but where the potential energy involves curvature. This sounds like an abstract mathematical problem, but it actually originates in the analysis of metallic regions with two different stable phases, when one tries to find the shapes of the regions with one or other phase. The second topic looks at certain purely algebraic identities recently discovered by Harrell and Stubbe, which turn out to apply to one of the central functions of statistical mechanics. The third topic has to do with the differences between the theory of quantum mechanics and the theory of classical physics, which applies to our world on a larger scale. Harrell has worked in this area for several years; new innovations include incorporating magnetism. The final topic has to do with some poorly understood local properties of quantum mechanical wavefunctions, and how measuring them might establish what sorts of forces are present. The methods used come from both pure and applied mathematics, and physics.