Spectral Analysis on Riemannian Manifolds

本文讨论了紧黎曼流形和开黎曼流形的谱分析问题。在紧流形上，一个基本的问题是：黎曼流形的几何在多大程度上是用它们的拉普拉斯谱编码的？这是一个在文献中被广泛研究的问题。PI是对具有不同局部几何形状的流形进行光谱研究的发起者之一。在PI构造的等谱例子中，最令人惊讶的是包含齐次度量和局部非齐次度量的等谱族。虽然这一领域在过去的15年里得到了很大的发展，但它远不是一个封闭的领域。到目前为止所做的所有构造都是关于函数谱的，对形式谱一无所知。例如，到目前为止，还没有已知的具有不同局部几何形状的度量，这些几何形状也是等谱的。将等谱研究扩展到各种形式，并最终解决这一长期存在的寻找具有不同局部几何形状的p-等谱紧致流形的问题，是本文的主要目标。这个目标是由PI最近发现的新的显式缠结算子实现的。这项新技术也为函数和形式的显式谱计算提供了可能性。开流形的谱分析是一个新兴的数学领域。由于基本核（如热核或薛定谔核）在开放流形上的无限迹，所有用于紧化流形的方法和工具在非紧化情况下都失效了。今天处理这些无穷大的唯一工具是所谓的“正则化”，通过这种方法，期望的有限数量由无穷大的差异产生。这种微扰工具是从量子理论（“重整化”）中借用来的，在量子理论中，上述无限大表现为粒子的无限自质量或自电荷。就在最近，PI有了一个新的非摄动工具，通过这个工具，这些无穷可以在相当大范围的黎曼流形上处理，叫做塞曼流形。通过在等谱构造中发展的显式谱计算，将塞曼流形上函数的希尔伯特空间分解为在拉普拉斯表示和自然海森堡群表示作用下不变的子空间（区）。因此，该算子可以在每个区域上单独研究，这意味着可以在每个区域上分别引入热流、薛定谔流、配分函数和ζ函数等重要对象。换句话说，可以开发一个定义良好的带状几何（德布罗意几何），其中最令人惊讶的结果是，在整体环境上的发散量在带状环境上是有限的。甚至区域费曼积分也有很好的定义。由于塞曼流形上的拉普拉斯算子只不过是自由带电粒子的塞曼-汉密尔顿算子，因此这些研究与量子物理学最为相关。本提案的主要焦点将是黎曼流形谱理论的一些基本问题。在紧形流形上，这个领域也可以被称为可听几何与不可听几何，这个名称很容易提出这个领域的基本问题：几何在多大程度上被编码到黎曼流形的拉普拉斯（谱）特征值中？大量的例子表明，光谱只包含很少的几何信息。然而，直到90年代初，人们普遍认为光谱确实决定了局部几何形状。PI是最早推翻这一猜想的人之一，并开始了对具有不同局部几何形状的流形的光谱研究。在PI构造的例子中，最有趣的是等谱度量，其中一个是齐次的（具有大量的等距在流形上传递），而另一个是局部非齐次的（只有“薄”组的局部等距，它们不作用于流形上传递）。这些例子表明，最重要的几何数据之一，等距线组，不是光谱确定的。虽然这一领域在过去的15年中发展很快，但这一理论还没有推广到形式。这一理论的扩展，包括解决长期存在的难题，即寻找局部非等距的p-等谱度量，是本提案的主要目标之一。非紧化流形的谱理论在计算自然核（如热核）的轨迹时遇到了无穷问题。由于这些无穷大，所有这些方法在非紧化流形上都失效了它们在紧化情况下是完美工作的。这个无穷大问题与出现在量子论中的无穷大问题是平行的。在这两种情况下，问题都是由一个微扰装置（正则化）处理的。重整化)，通过无穷差产生期望的有限量。作为第二个主题，本文提出了一种新的、自然的、非摄动的装置，它可以直接在非紧流形上产生期望的有限量。这个工具也与量子理论有关。

This proposal is concerned with spectral analysis both on compact and open Riemannian manifolds. On compact manifolds, one of the fundamental questions is: To what extent is the geometry of Riemannian manifolds encoded in the spectra of their Laplacians? This is a very extensively investigated question in the literature. The PI is one of the initiators of spectral investigations developed on manifolds having different local geometries. Among the isospectrality examples the PI constructed the most surprising are the isospectrality families containing both homogeneous and locally inhomogeneous metrics. Although this field became very developed in the past 15 years, it is far from being a closed area. All the constructions performed so far deal with the function-spectra and nothing is known about the form-spectra. For instance, no metrics with different local geometries are known, upto this day, whichare isospectral also on forms. The extension of the isospectrality investigations to the forms and the ultimate solution of this long standing problem of finding p-isospectral compact manifolds with different local geometries belong to the main objectives of this proposal. This goal is aimed by the new explicit intertwining operator found by the PI just recently. This new technique opens up the possibility also for explicit spectrum computations, both on functions and forms. Spectral analysis on open manifols is a developing area of mathematics. Because of the infinite trace of the fundamental kernels (such as the heat-or Schroedinger-kernel) on the open manifolds, all the methods and tools applied on compact manifolds break down in the non-compact cases. The only tool by which these infinities are handled today is the so called "regularization" by which the desired finite quantities are produced by differences of infinities. This perturbative tool has been borrowed from quantum theory ("renormalization"), where the above infinities appear as infinite self mass or self charge of particles. Just recently, the PI has a new non-pertubative tool found by which these infinities can be handled on a rather wide range of Riemannian manifolds, called Zeeman manifolds. By the explicit spectrum computation, developed in isospectrality constructions, the Hilbert space of functions on a Zeeman manifold decomposes into subspaces (zones) which are invariant under the actions of the Laplacian and the natural Heisenberg group representation. Therefore, this operator can be investigated on each zone separately, meaning that important objects such as the heat flow, Schroedinger flow, partition- and zeta-function, e. t. c. can be introduced on each zone separately. In other words, a well defined zonal geometry (de Broglie geometry) can be developed, where the most surprising result is that quantities divergent on the global setting are finite on the zonal setting. Even the zonal Feynman integral is well defined. Since the Laplacian on Zeeman manifolds is nothing but the Zeeman-Hamilton operators of free charged particles, these investigations are most relevant to the quantum physics.The main focus of this proposal will be some of the fundamental questions of spectral theory on Riemannian manifolds. On compact manifolds this field might as well be called audible versus nonaudible geometry, which designation readily suggests the fundamental question of the area: To what extend is the geometry encoded into the eigenvalues of the Laplacian (spectrum) of a Riemannian manifold? A wide range of examples show that the spectrum bears just little information about the geometry. Yet, until the early 90's, a general expectation was that the spectrum does determine the local geometry. The PI was one of the first ones who disproved this conjecture and initiated spectral investigations on manifolds having different local geometries. Among the examples the PI constructed the most interesting are the isospectral metrics such that one of them is homogeneous (having a huge group of isometries acting transitively on the manifold) while the other is locally inhomogeneous (having just a "thin" group of local isometries which do not act transitively on the manifold). These examples show that one of the most important geometric data, the group of isometries, is not spectrally determined. Though this field developed very rapidly in the past 15 years, this theory has not been extended to the forms yet. This extension of the theory, including also the solution of the long standing difficult problem of finding locally non-isometric yet p-isospectral metrics, is one of the main objectives of this proposal. Spectral theory on non-compact manifolds is struggling with the infinities appearing in calculating the trace of natural kernels such as the heat kernel. Due to these infinities, all those methods break down on non-compact manifolds which are perfectly working in the compact case. This problem of infinities is parallel to the problem of infinities appearing in quantum theory. In both cases the problem is handled by a perturbative device (regularization resp. renormalization), producing the desired finite quantities by differences of infinities. As a second subject, this proposal introduces a new, natural, non-perturbative device which produces the desired finite quantities on non-compact manifolds directly. This tool is relevant also to quantum theory.