Spectral and Asymptotoc Problems of Mathematical Physics

摘要Harrell 9622730 本计画系关于在量子物理与反应扩散中所产生的微分算子的光谱理论中的几个问题的数学分析。 人们希望这些结果对物理学家有真实的用处，同时又能用于严格的原始泛函分析或微分方程理论。 许多特殊的问题与共振和亚稳性有关，或者与几何、算子代数和光谱之间的联系有关。 第一个具体的主题在该提案涉及的特征值薛定谔运营商的势能涉及曲率。 这听起来很抽象，但它实际上起源于对双金属合金相区形状的分析。 第二个主题着眼于某些纯粹的代数身份最近发现的哈雷尔和Stubbe，这原来适用于，急剧，分区函数（=跟踪的热核），以及其他谱表达式。 第三个主题与半经典分析有关，Harrell在这个领域工作了几年，但新的特征包括磁场和时间相关的相互作用。 最后一个主题涉及本征函数的局部性质，如它们的节点，以及一些涉及它们的逆问题。 数学机器被称为包括扰动理论，渐近，变分分析。 这个项目是关于物理学中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。 许多特殊的问题与共振和亚稳态（缓慢变化）现象有关，其他问题与几何在这些物理领域中的作用有关。 研究的目的是要产生一些新的和有用的工作物理学家，同时阐明数学结构的物理理论所依赖的第一个具体的主题是作出估计的“正常模式” 类似于量子力学中的方程，但势能涉及曲率。 这听起来像是一个抽象的数学问题，但它实际上起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。 第二个主题着眼于某些纯粹的代数身份最近发现的哈雷尔和Stubbe，这原来适用于中央职能之一的统计力学。 第三个主题是关于 量子力学理论和经典物理理论之间的差异， 这在更大的范围内适用于我们的世界。 Harrell已经在这一领域工作了好几年;新的创新包括将磁性。 最后一个主题是关于量子力学波函数的一些知之甚少的局部性质，以及如何测量它们来确定存在什么样的力。 所使用的方法来自纯数学和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.