Spectral and geometric problems in global analysis

全局分析中的谱和几何问题

• 批准号: 0605247
• 负责人: Carolyn Gordon
• 金额: $ 20.95万
• 依托单位: Dartmouth College
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2010-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0605247&HistoricalAwards=false
• 关键词: Spectral; geometric; problems; global; analysis

Inverse spectral geometry is the study of the extent to which the geometry of a Riemannian manifold can be recovered from spectral data. The principal investigator, along with her research collaborators, will study Inverse spectral problems in both the compact and noncompact settings. For compact Riemannian manifolds, the natural spectral data are the eigenvalues of the Laplacian. To obtain information concerning Riemannian invariants that are not spectrally determined, constructions of isospectral manifolds will be investigated. Continuing joint work with E. Makover and D. Webb, the principal investigator will study the extent to which the spectrum of a Riemann surface determines its Jacobian. She and Webb, along with E. Dryden and S. Greenwald, will also consider inverse spectral results on orbifolds. For noncompact manifolds, natural spectral data include the scattering resonances and scattering phase. The principal investigator and Webb, along with P. Perry, will study of obstacles with the same scattering resonances and scattering phase. They will also study isopolar and isophasal potentials for the Schrodinger operator.In spectroscopy, one attempts to understand the chemical composition of an object such as a star from the characteristic frequencies of light emitted. Analogously, the question phrased by Mark Kac as "Can one hear the shape of a drum?" asks whether one can determine the shape of a vibrating membrane such as a drumhead from its characteristic frequencies of vibration. In earlier work, the principal investigator, along with D. Webb and S. Wolpert, constructed examples of polygonal shaped "membranes" (bounded domains in the plane) that have exactly the same characteristic frequencies, thus answering Kac's question in the negative. Kac's question generalizes to nonplanar surfaces and higher dimensional objects (Riemannian manifolds), with the analog of the characteristic frequencies being the Laplace spectrum. The principal investigator, along with her collaborators, will continue her investigation of the extent to which spectral data determines the geometry of a Riemannian manifold. Constructions of manifolds with the same spectrum will be studied to identify geometric properties that are not spectrally determined. Singular spaces (orbifolds) will also be considered.

逆谱几何是研究从谱数据恢复黎曼流形几何形状的程度。 首席研究员和她的研究合作者将研究紧凑和非紧凑环境中的逆谱问题。 对于紧黎曼流形，自然光谱数据是拉普拉斯算子的特征值。 为了获得有关非谱确定的黎曼不变量的信息，将研究等谱流形的构造。 首席研究员将继续与 E. Makover 和 D. Webb 合作，研究黎曼曲面的谱在多大程度上决定其雅可比行列式。 她和韦伯以及 E. Dryden 和 S. Greenwald 还将考虑轨道折叠的逆光谱结果。 对于非紧流形，自然光谱数据包括散射共振和散射相位。 首席研究员和韦伯以及 P. Perry 将研究具有相同散射共振和散射相位的障碍物。 他们还将研究薛定谔算子的等极和等相势。在光谱学中，人们试图从发射光的特征频率来了解物体（例如恒星）的化学成分。 类似地，马克·卡克（Mark Kac）提出的问题是“人们能听到鼓的形状吗？”询问是否可以根据振动的特征频率确定振动膜（例如鼓面）的形状。 在早期的工作中，首席研究员与 D. Webb 和 S. Wolpert 一起构建了具有完全相同特征频率的多边形“膜”（平面上的有界域）的例子，从而对 Kac 的问题给出了否定的回答。 Kac 的问题推广到非平面表面和高维物体（黎曼流形），特征频率的模拟是拉普拉斯谱。 首席研究员和她的合作者将继续研究光谱数据在多大程度上决定黎曼流形的几何形状。 将研究具有相同光谱的流形的构造，以识别未由光谱确定的几何特性。 奇异空间（orbifolds）也将被考虑。