Mathematical Sciences: The Spectral Theory of Elliptic Operators: Microlocal Approaches

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The principal investigator will study the point spectra of elliptic self-adjoint (pseudodifferential) operators in a variety of contexts. This study is fundamental to the understanding of the relationship between analysis and geometry, and is of interest in mathematical physics. The project will emphasize the point of view of microlocal analysis. A microlocal version of the Selberg trace formula for elliptic operators will be pursued. Results from this project will then be applied to inverse spectral problems. The work to be completed in this project will extend our knowledge of the relationship between differential operators and geometry. In this century mathematical physics has seen the great rise of operator theory as a tool for understanding the quantum mechanics of subatomic particles. The principal investigator will develop microlocal analytic techniques to solve various fundamental problems concerning these important operators.