Mathematical Sciences: Spectral and Symptotic Problems of Mathematical Physics

哈雷尔将研究量子力学的隧道效应和方程的间歇性，例如激光物理学中出现的方程。 其他问题涉及几何和分析之间的相互作用，以及试图理解非线性现象。 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，在应用中经常出现超出既定理论范围的奇异算子或限制，并激发了该项目的大部分内容。 现代物理学、量子力学和相对论，是二十世纪的产物。 它牢固地建立在上个世纪解决物质微观结构并掌握远距离作用、电磁和热辐射概念的尝试中。 这些发展的数学基础统称为数学物理，范围从控制粒子动力学的薛定谔算子的详细分析，到试图将四种已知力统一为一个理论的统一场论。

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.