Spectral Geometry of Non-Compact Domains and Riemannian Manifolds

摘要对于DMS-0100829，主要研究人员将研究非紧域和黎曼流形的谱几何，以阐明散射极点的几何内容。首先，PI将继续他对双曲流形的谱几何及其扰动的研究。他将研究作为下面离散群形变空间上的函数的共振，并定义和分析拉普拉斯行列式。其次，PI将研究两步幂零李群上的波动方程及其离散子群的商的散射理论。将推导出海森堡型和海森堡型群上的新参数或波动方程，并获得某些商的迹公式。将使用Gordon、Wilson等人的黎曼淹没技术来获得成对或成族的等散射流形，这将有助于确定几何信息的限度，这些信息可以从散射极点的知识中推导出来。第三，PI将研究欧氏空间中外部区域的等散射问题。谱几何的基本问题是阐明黎曼流形上拉普拉斯谱的几何内容。对于所谓的散射流形，拉普拉斯谱的本征值与散射共振一起构成流形的谱数据。阐明这类光谱数据的几何内容有助于我们理解量子化，为几何对象的研究提供新的分析工具，并提供对本征值和散射极点是可测量的量的更“实用”性质的逆问题的洞察。本工作的目的是从可以使用李氏理论、自同构函数和调和分析技术的几何自然例子开始，并进展到诸如雷达目标识别之类的困难问题，在这些问题中，这些技术不可用，但基本的数学问题非常相似。

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.