Differential Operators in Several Complex Variables

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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.