Spectral Analysis on Riemannian Manifolds

这个建议涉及紧致黎曼流形和开黎曼流形上的谱分析。在紧致流形上，一个基本问题是：黎曼流形的几何在多大程度上编码在其拉普拉斯流形的谱中？这是一个在文献中被广泛调查的问题。PI是在具有不同局部几何结构的流形上发展起来的光谱研究的发起者之一。在所构造的等谱族中，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年里变得非常发达，但它远远不是一个封闭的区域。到目前为止，所有的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，还没有已知具有不同局部几何的度量，这些度量在形式上也是等谱的。将等谱研究推广到形式和寻找具有不同局部几何的p-等谱紧致流形这一长期存在的问题的最终解是本提议的主要目的。PI最近发现的一种新的显式交织算子就是为了达到这个目标。这一新技术也为函数和形式上的显式谱计算打开了可能性。开流形上的谱分析是一个发展中的数学领域。由于基本核(如热核或薛定谔核)在开流形上的无限迹，所有应用于紧致流形的方法和工具在非紧致情况下都失效了。今天，处理这些无穷大的唯一工具是所谓的“正则化”，即通过无穷大的差异产生所需的有限数量。这一微扰工具是从量子理论(“重整化”)中借来的，在量子理论中，上述无穷大表现为粒子的无穷大自质量或自电荷。就在最近，PI发现了一种新的非摄动工具，通过它可以在相当广泛的黎曼流形上处理这些无穷大，称为塞曼流形。通过在等谱结构中发展的显式谱计算，塞曼流形上的函数的希尔伯特空间被分解成在拉普拉斯和自然海森堡群表示的作用下不变的子空间(带)。因此，这个算符可以在每个区域上单独研究，这意味着重要的对象，如热流、薛定谔流、配分函数和Zeta函数等，可以分别在每个区域上引入。换句话说，可以开发一个定义良好的分区几何(德布罗意几何)，其中最令人惊讶的结果是在全局设置上发散的量在分区设置上是有限的。即使是带状费曼积分也是定义良好的。由于塞曼流形上的拉普拉斯算符只不过是自由带电粒子的塞曼-哈密尔顿算符，因此这些研究与量子物理最为相关。这一提议的主要焦点将是黎曼流形上的谱理论的一些基本问题。在紧致流形上，这个场也可以被称为可听几何和不可听几何，这个名称很容易地提出这个区域的基本问题：几何在多大程度上被编码到黎曼流形的拉普拉斯(谱)的特征值中？大量的例子表明，光谱包含的几何信息很少。然而，直到90年代初的S，一个普遍的期望是光谱确实决定了局部几何。PI是第一批驳斥这一猜想的人之一，并开始了对具有不同局部几何的流形的谱研究。在PI构造的例子中，最有趣的是等谱度量，使得其中一个是齐次的(有一大组等距线在流形上传递地作用)，而另一个是局部非齐次的(只有一组不在流形上传递作用的局部等距线组)。这些例子表明，最重要的几何数据之一等轴测线组并不是光谱确定的。虽然这一领域在过去的15年里发展很快，但这一理论还没有扩展到形式上。这一理论的扩展，包括寻找局部非等距但p等谱度量这一长期存在的难题的解决，是这一提议的主要目标之一。非紧致流形上的谱理论正在与计算自然核(如热核)的迹时出现的无穷大作斗争。由于这些无穷大，所有这些方法在非紧致流形上都失效了，而这些流形在紧致情况下是完美工作的。这个无穷大问题与量子理论中出现的无穷大问题是平行的。在这两种情况下，问题都是由微扰装置处理的(正则化对应。重整化)，通过无穷大的差异产生所需的有限量。作为第二个主题，该建议引入了一种新的、自然的、非微扰的装置，它直接在非紧致流形上产生所需的有限量。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.