Mathematical Sciences: Spectral and Symptotic Problems of Mathematical Physics

哈雷尔将致力于量子的隧道效应 力学和方程中的不相容性， 在激光物理学中。 其他问题涉及到 几何和分析，并试图了解非线性 现象。 谱理论是许多物理问题的核心; 例如，量子力学中的能级被确定为 某些线性算子的特征值。 然而，奇异的 操作员或界限不在已确立的 理论经常出现在应用程序中，并激发了这一点。 项目 现代物理学，量子力学和相对论，是一个 二十世纪的产物。 它牢固地建立在 上世纪人们试图研究物质的微观结构 并掌握远距离行动的概念， 电磁和热辐射。 数学 这些发展的基础，统称为 数学物理，范围从详细分析 薛定谔算子，控制着粒子的动力学， 统一场论，试图统一四个已知的 形成一个单一的理论。

Harrell will work on the tunneling effect of quantum mechanics and on intermittency in equations such as those arising in laser physics. Other problems involve the interplay between geometry and analysis, and with attempts to understand nonlinear phenomena. Spectral theory is central to many physical problems; for example energy levels in quantum mechanics are determined as eigenvalues of certain linear operators. However, singular operators or limits lying outside the scope of well-established theories often arise in applications, and motivate much of this project. Modern physics, quantum mechanics and relativity, is a product of the twentieth century. It is founded firmly in the last century's attempt to address the microstructure of matter and to come to grips with the concepts of action-at-a distance, electro-magnetism, and heat radiation. The mathematical foundations for these developments, collectively called mathematical physics, ranges from detailed analysis of Schroedinger operators, which governs the dynamics of particles, to unified field theory, which attempts to unite the four known forces into a single theory.