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）厄米全纯线丛上拉普拉斯算子的本征值估计，ii）量子唯一遍历性，iii）双曲空间理想边界附近本征函数的行为。考虑紧致复流形上的埃尔米特全纯线丛的张量幂。如果曲率形式的线丛是严格积极的，那么第一个非零特征值，拉普拉斯算子，作用于部分的$k$次幂，是有界的一致低于$k$。本文的目的是证明当曲率处处半正且至少对一点为正时，第一非零特征值在下一致有界。这个问题可以用CR几何来重新表述。微局部分析将被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本征函数的集中。我们建议构造的例子中，本征函数的序列集中沿着孤立的封闭测地线或封闭测地线的一个参数的家庭。第一步是找到准模（近似本征函数）。其次，必须证明这些准模对应于个别本征函数，而不是几个本征函数的总和。双曲空间存在本质谱，它是正真实的直线的真子集。确实存在定义在紧集的补集上的本征函数，其本征值位于本质谱的起始点以下。我们将研究这些本征函数在无穷远理想边界附近的行为。我们的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。为了发展我们对量子现象的理解，数学家们经常受到与经典力学类比的启发。从经典能级到量子能级的过渡称为半经典极限。如果经典运动是混沌的，那么量子粒子的概率分布就会被分散。在这种模式下，颗粒集中，是特别感兴趣的。类似地，如果能量估计在经典曲率形式的严格正性条件下成立，人们自然会研究曲率形式为非负的边界情况。人们希望正则性性质在更一般的情况下保持不变。这类问题很有趣，因为非负对象经常作为正对象的极限出现。一个量子粒子的节集是其波运动的定态集。它的分布和形状是基本的兴趣，但知之甚少，特别是在大于2的维度。

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.