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

主要研究者将研究椭圆型自伴随（伪微分）算子在各种情况下的点谱。这项研究是理解分析和几何之间关系的基础，也是数学物理的兴趣所在。本项目将强调微观局部分析的观点。将讨论椭圆算子Selberg迹公式的微局部版本。这个项目的结果将应用于反光谱问题。在这个项目中完成的工作将扩展我们对微分算子和几何之间关系的认识。在本世纪，数学物理学见证了作为理解亚原子粒子量子力学工具的算符理论的巨大崛起。首席研究员将开发微局部分析技术来解决有关这些重要操作符的各种基本问题。

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.