Spectral Geometry of Non-Compact Domains and Riemannian Manifolds

非紧域和黎曼流形的谱几何

• 批准号: 0100829
• 负责人: Peter Perry
• 金额: $ 9万
• 依托单位: University of Kentucky
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-07-01 至 2005-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0100829&HistoricalAwards=false
• 关键词: Spectral; Geometry; Non; Compact; Domains

Abstract for DMS - 0100829The principal investigator will study the spectral geometry of non-compactdomains and Riemannian manifolds in order to elucidate the geometriccontent of scattering poles. First, the PI will continue his study of thespectral geometry of hyperbolic manifolds and their perturbations. He willstudy resonances as functions on the deformation space of the underlyingdiscrete group, and define and analyze a determinant of the Laplacian.Secondly, the PI will study scattering theory for the wave equation ontwo-step nilpotent Lie groups and their quotients by discrete subgroups.New parametrices or the wave equation on the Heisenberg and Heisenberg-typegroups will be derived, and trace formula for certain quotients obtained.Riemannian submersion techniques of Gordon, Wilson, and others will be usedto obtain pairs and families of `isoscattering' manifolds which will helpdetermine the limits of geometric information which may be deduced from aknowledge of the scattering poles. Thirdly, the PI will study theisoscattering problem for exterior domains in Euclidean space.The fundamental problem of spectral geometry is to elucidate thegeometric content of the Laplace spectrum on a Riemannian manifold.For so-called scattering manifolds, the eigenvalues of the Laplaciantogether with scattering resonances constitute the spectral data for themanifold. Elucidating the geometric content of such spectral dataadvances our understanding of quantization, produces new analytic toolsfor the study of geometric objects, and provides insight into inverseproblems of a more `applied' nature where the eigenvalues and scatteringpoles are measurable quantities. The present work aims to begin withgeometrically natural examples where techniques of Lie theory, automorphicfunctions, and harmonic analysis may be used, and progress to harderproblems such as target identification by radar where such techniques are not available but the underlying mathematical problems are very similar.

DMS -0100829的摘要主要研究者将研究非紧密含量和riemannian歧管的光谱几何形状，以阐明散射杆的几何表现。首先，PI将继续研究双曲线歧管及其扰动的几何形状。 He willstudy resonances as functions on the deformation space of the underlyingdiscrete group, and define and analyze a determinant of the Laplacian.Secondly, the PI will study scattering theory for the wave equation ontwo-step nilpotent Lie groups and their quotients by discrete subgroups.New parametrices or the wave equation on the Heisenberg and Heisenberg-typegroups将被得出的某些商。 Thirdly, the PI will study theisoscattering problem for exterior domains in Euclidean space.The fundamental problem of spectral geometry is to elucidate thegeometric content of the Laplace spectrum on a Riemannian manifold.For so-called scattering manifolds, the eigenvalues of the Laplaciantogether with scattering resonances constitute the spectral data for themanifold. 阐明这种频谱数据吸收的几何含量使我们对量化的理解，为研究几何对象的研究产生新的分析工具，并提供对更“应用”性质的逆问题的见识，在该逆问题中，特征值和散射柱是可衡量的数量。目前的工作旨在从拼小的自然示例开始，在这些示例中，可以使用谎言理论，自动化功能和谐波分析的技术，并在无法获得此类技术但基本数学问题的雷达中进行诸如目标识别之类的硬问题。