Spectral Theory of Riemannian Manifolds

摘要DMS - 0203070。主要研究者计划研究三个独立的主题：i)拉普拉斯在厄米全纯线束上的特征值估计，ii)量子唯一遍历性，以及iii)双曲空间理想边界附近特征函数的行为。考虑一个紧复流形上的厄密全纯线束的张量幂。如果线束的曲率形式是严格正的，那么作用于k次幂部分的拉普拉斯函数的第一个非零特征值，在k次幂上均匀地有界于下面。证明了当曲率处处为半正且至少有一点为正时，第一个非零特征值是一致有界的。这个问题可以用CR几何重新表述。微局部分析将应用于重新表述的问题。量子唯一遍历性问题涉及具有遍历测地线流的流形上特征函数的集中问题。我们提出构造特征函数序列沿孤立的封闭测地线或在封闭测地线的一个参数族上集中的例子。第一步是找到准模（近似特征函数）。接下来必须证明这些准模对应于单个特征函数而不是几个特征函数的和。双曲空间具有本质谱，本质谱是正实线的固有子集。确实存在定义在紧集补上的特征函数，其特征值位于本质谱的起始点以下。这些本征函数的性质将在无限远处的理想边界附近进行研究。目标是通过微扰展开来理解节点集。表面的情况似乎比高维的情况更容易处理。为了加深我们对量子现象的理解，数学家们经常从与经典力学的类比中得到启发。从经典到量子的过渡称为半经典极限。如果经典运动是混沌的，人们期望量子粒子的概率分布是分散的。这种模式的例外，即粒子集中的地方，是特别有趣的。类似地，如果能量估计在经典曲率形式的严格正条件下成立，人们自然会研究曲率形式非负的边界情况。人们希望规律性能在更一般的情况下保持不变。这类问题很有趣，因为非否定宾语经常作为肯定宾语的限制出现。量子粒子的节点集是相关波动的平稳集。它的分布和形状是人们最感兴趣的，但人们却知之甚少，尤其是在大于2的维度上。

ABSTRACT DMS - 0203070.The principal investigator plans to study three separate topics:i) Eigenvalue estimates for the Laplacian on Hermitian holomorphic line bundles, ii) Quantum Unique Ergodicity, and iii) Behavior of eigenfunctions near the ideal boundary of hyperbolic space. Consider the tensor powers of a Hermitian holomorphic line bundle, over a compact complex manifold. If the curvature form of the line bundle is strictly positive, then the first non--zero eigenvalue, of the Laplacian, acting on sections of the $k$th power, is bounded below uniformly in $k$. The proposal is to prove that the first non--zero eigenvalue is uniformly bounded below when the curvature is semipositive everywhere and positive for at least one point. The problem can be reformulatedin terms of CR geometry. Microlocal analysis will be applied to the reformulated problem.The problem of quantum unique ergodicity concerns concentration of eigenfunctions on manifolds with ergodic geodesic flow. We propose to construct examples where sequences of eigenfunctions concentrate along isolated closed geodesics or on one parameter families of closed geodesics. The first step is to find quasimodes (approximate eigenfunctions). Next one must show that these quasimodes correspond to individual eigenfunctions rather than sums of several eigenfunctions. The hyperbolic space has essential spectrum which is a proper subset of the positive real line. There do exist eigenfunctions, defined on the complements of compact sets, whose eigenvalue lies below the start of the essential spectrum.The behavior of these eigenfunctions will be studied near the ideal boundary at infinity.The goal is to understand the nodal set by means of a perturbation expansion. It appears that the case of surfaces is much more tractable than the higher dimensional cases.To develop our understanding of the quantum phenomena, mathematicians are often inspired by the analogy with classical mechanics. The passage from the classical to the quantum level is called the semiclassical limit. If the classical motion is chaotic, one expects the probability distribution of the quantum particle to be dispersed. Exceptions to this pattern, where the particle concentrates, are of particular interest. Similarly, if an energy estimate holds under strict positivity conditions of a classical curvature form, one naturally investigates the borderline case where the curvature form is non--negative.One hopes that regularity properties will persevere in the more general situation.This type of question is interesting because non--negative objects often occur as the limitsof positive objects. The nodal set of a quantum particle is the stationary set for the associated wave motion. Its distribution and shape are of fundamental interest, but poorly understood, especially in dimensions larger than two.