Inverse Problems and Spectral Theory for Elliptic Operators

该项目涉及科学技术中出现的几个基本反问题的数学理论的发展，以及在来自电磁和量子力学问题的谱理论的重要问题上取得进展。 从广义上讲，在逆问题中，人们希望通过沿介质的边界进行测量来确定介质的内部属性。例如，在电阻抗断层扫描中，人们试图通过在边界处进行电压和电流测量来恢复身体的电导率。 由于反问题是各种工程和科学研究的核心，包括医学成像、地震学、石油勘探、雷达成像和无损检测，因此此类问题的数学理论的任何进一步进展无疑将具有现实世界的应用。 谱理论涉及各种不同物体的振动及其频率的研究，从化学中的原子和分子到声波导中的障碍物。基本问题在从天体到量子力学的许多科学和工程问题中具有重要意义，包括确定这种振动何时发生、如何计算其频率以及了解振动的大小和位置。该项目的目的是通过专注于量子力学的模型问题，特别是在高频的物理相关机制中，增进我们对这些问题的理解。该项目讨论以下重要主题：椭圆偏微分方程 (PDE) 的逆边界问题的数学理论、椭圆 PDE 的调和分析以及具有周期系数的椭圆 PDE 的谱理论。尽管这些主题起源于不同的数学社区，但最近的工作表明，各个主题的技术和见解密切相关，并以富有成效的方式相互作用。该项目的一个新颖想法是扩大这种互动，以解决所有这些领域的重大问题。 尽管过去 30 年来在反演问题领域取得了令人印象深刻的成果，但许多基本问题仍未得到解决，包括具有不规则系数的偏微分方程的反演问题、仅在部分边界上进行测量时的部分数据问题以及流形上的反演问题。该项目第一部分的目标是解决几个基本椭圆偏微分方程的这些问题，特别是电导率、磁薛定谔方程、多调和方程以及麦克斯韦系统。第二个主题涉及勒贝格空间中紧致流形和非紧流形上椭圆算子的解析解的估计。除了它们在光谱和散射理论中的内在意义之外，这种估计在控制理论和反演问题中也至关重要。这里的目的是了解算子的基础汉密尔顿流的动态性和系数的规律性如何影响谱估计。第三个主题涉及来自固体物理学的具有周期系数的薛定谔型算子的谱理论。一个核心问题涉及此类算子的谱的性质，人们推测其谱是纯粹绝对连续的。 欧几里得案例早已为人所知，但它对于一般的 Laplace-Beltrami 类型算子仍然是开放的。目标是在黎曼度量的几个重要特例中解决这个猜想。

The project is concerned with development of the mathematical theory for several fundamental inverse problems arising in science and technology, as well as with achieving advances in significant questions of spectral theory coming from problems of electromagnetism and quantum mechanics. Broadly speaking, in an inverse problem, one wishes to determine internal properties of a medium by performing measurements along the boundary of the medium. For instance, in electrical impedance tomography, one attempts to recover the conductivity of a body by making voltage and current measurements at the boundary. Since inverse problems are at the core of a variety of engineering and scientific investigations, including medical imaging, seismography, oil prospection, radar imaging, and non-destructive testing, any further progress in the mathematical theory of such problems will undoubtedly have real world applications. Spectral theory deals with the investigation of vibrations and their frequencies for a variety of different objects, ranging from atoms and molecules in chemistry to obstacles in acoustic waveguides. The fundamental issues, which are of great significance in many problems of science and engineering, from celestial to quantum mechanics, include deciding when such vibrations occur, how to go about computing their frequencies, as well as understanding the size and localization of the vibrations. The aim of the project is to advance our understanding of these issues by concentrating on model problems of quantum mechanics, specifically in the physically relevant regime of high frequencies.The project addresses the following significant topics: the mathematical theory of inverse boundary problems for elliptic partial differential equations (PDE), harmonic analysis for elliptic PDE, and spectral theory of elliptic PDE with periodic coefficients. Although these topics have originated in distinct mathematical communities, recent work has shown that techniques and insights in the various topics are closely related and interact in a fruitful way. A novel idea of the project is to expand this interaction to solve significant problems in all of these areas. Despite an impressive body of results in the field of inverse problems obtained within the last 30 years, many fundamental questions still remain unsolved, including inverse problems for PDE with irregular coefficients, partial data problems when measurements are performed only on a portion of the boundary, and inverse problems on manifolds. The goal of the first part of the project is to attack these problems for several fundamental elliptic PDE, in particular the conductivity, magnetic Schroedinger, and polyharmonic equations, as well as the Maxwell system. The second topic is concerned with estimates for resolvents of elliptic operators on compact and non-compact manifolds, in Lebesgue spaces. Apart from their intrinsic significance in spectral and scattering theory, such estimates are crucial in control theory and inverse problems. The aim here is to understand how the dynamics of the underlying Hamilton flow of the operator and regularity of the coefficients each impacts on the spectral estimates. The third topic deals with the spectral theory of Schroedinger type operators with periodic coefficients, coming from solid state physics. A central question concerns the nature of the spectra of such operators, which one conjectures to be purely absolutely continuous. Long known in the Euclidean case, it is still wide open for general Laplace-Beltrami type operators. The objective is to resolve this conjecture in several significant special cases of Riemannian metrics.