Differential Operators in Several Complex Variables

摘要本课题研究复变量的复数和Kohn laplacian，复流形上的d-bar问题，以及它们在代数几何和量子物理问题中的应用和相互作用。提出的研究的主要重点之一是沿着狄利克雷和诺伊曼拉普拉斯的经典谱理论对d-bar-Neumann和kohn拉普拉斯进行系统的研究。研究者将继续研究这些拉普拉斯子的光谱行为与底层几何结构之间的相互作用。一个特别有趣的问题是描述这些拉普拉斯算子具有纯离散谱的几何结构。研究者还将研究与磁场相关的薛定谔算符的谱理论。复杂的分析和微分方程在物理学、工程学、化学、经济学和其他科学中自然出现。微分算子的谱分析在量子力学中起着重要的作用。提出的研究将进一步探索几个复杂变量中看似不相关的问题与量子力学之间的密切联系。它可能会在这两个领域带来新的认识和发现。在这个项目中开发的想法、工具和技术也将在其他领域产生影响。这个项目将对人力资源的发展产生直接影响；它将支持开发新的课程和教材，吸引学生进入数学和科学领域。它将促进生物学家和计算机科学家的跨学科研究活动。

ABSTRACTThis project studies the complex and Kohn Laplacians in severalcomplex variables, the d-bar-problem on complex manifolds, as wellas their applications to and interplays with problems in algebraicgeometry and quantum physics. One of the main thrusts of theproposed research is systematical study of the d-bar-Neumann andKohn Laplacians along the lines of the classical spectral theoryfor the Dirichlet and Neumann Laplacians. The investigator willcontinue his work on the interplays between spectral behavior ofthese Laplacians and the underline geometric structures. A problemof particular interest is to characterize geometric structures onwhich these Laplacians have purely discrete spectrum. Theinvestigator will also study the spectral theory of relatedSchrodinger operators with magnetic fields.Complex analysis and differential equations arise naturally inphysics, engineering, chemistry, economics, and other sciences.Spectral analysis of differential operators plays an importantrole in quantum mechanics. The proposed research will furtherexplore the intimate connections between the seemingly unrelatedproblems in several complex variables and quantum mechanics. Itcould lead to new understanding and discoveries in both fields.Ideas, tools, and techniques developed in this project will alsohave repercussions in other fields. This project will have directimpact on the development of human resources; it will supportdevelopment of new courses and texts that attract students intomathematics and sciences. It will facilitate interdisciplinaryresearch activities with biologists and computer scientists.