Asymptotic Geometry of Eigenfunctions and Polynomials

摘要：本文讨论了黎曼流形上拉普拉斯算子的特征函数和特征值的渐近性，并对Kahler流形上的复拉普拉斯算子进行了类比。 利用半经典分析方法（量子力学经典极限的数学），本征函数的高频极限与测地线流的动力学有关。 第一个应用是著名的解析平面区域的逆谱问题，新的思想是使用Dirichlet预解式的精确公式和它的迹的Feynman图分析。首次得到了用弹跳球轨道的域端点的泰勒系数表示的所谓波迹不变量的显式公式。从这些人试图恢复泰勒系数，因此一个解析域。 迄今为止，最好的结果是可以从它的谱中恢复出一个镜像对称的解析域。第二个应用是本征函数的几何。Sogge和提出者证明了特征函数的上范数的最大可能增长率只出现在包含回归点的无边界黎曼流形上，使得离开回归点的测地线在固定时间内回归。 一个项目是证明这一点（如果真的）有界域。类似的分析对于本征函数的边界值也是有趣的，这在边界控制理论中是很重要的。Hassell和提出者最近证明了当域的台球流是遍历时本征函数边界值的遍历性，并给出了边界值L2模的第一个渐近估计。 本文所发展的方法对本征函数的几何学有进一步的应用。第三个项目，与B联合。Shiffmanis应用半经典方法统计代数几何，即，to finding发现statistical统计patterns模式in the zeroes零点of polynomials多项式. 在迄今发现的模式中，多项式系统的零点往往在一维中排斥，在二维中像中性气体一样，在高维中吸引。新的问题是探索零的分布对牛顿多面体或单项式的数量的依赖性，如在Khovanski的fewnomial理论中。半经典分析是数学领域，探索量子力学的经典极限，即原子和分子的模糊，跳跃的小尺度行为与我们所经历的固体，机械世界合并的极限。普朗克常数h测量长度尺度。虽然它起源于物理学，但同样的极限，当h趋于零时，也出现在数学和科学的许多问题中，往往与最初的物理背景完全脱节。这个建议涉及到几个这类问题。有几个是传统的和众所周知的问题，例如，试图确定一个域的振动频率。 使用半经典分析的一种更熟悉的拓扑学家比数学家（Balian-Bloch方法，费曼图），第一个项目是获得最好的结果，到目前为止的问题，可以听到鼓的形状。 到目前为止的结果表明，这种方法是优于任何现有的方法。另一个项目是观察非多项式1/h（逆普朗克常数）的次数，并使用半经典方法找到多项式零点和临界点的新统计模式。 结果之一是多项式中出现的指数对零点的位置有很大的影响，导致了类似于电子穿过势垒的隧穿理论的零点隧穿理论。

ASYMPTOTIC GEOMETRY OF EIGENFUNCTIONS AND POLYNOMIALSAbstract:This proposal is concerned with the asymptotics of eigenfunctions and eigenvalues of the Laplacian on Riemannian manifold, and analoguesfor the complex Laplacian on Kahler manifolds. Using methods ofsemi-classical analysis (the mathematics of the classical limit ofquantum mechanics), the high frequency limit of eigenfunctions is relatedto the dynamics of the geodesic flow. The first application is to thewell-known inverse spectral problem for analytic plane domains.The new idea is to use an exactformula for the Dirichlet resolvent and a Feynmandiagram analysis of its trace. One obtains for the first time explicit formulae for so-calledwave trace invariants in terms of the Taylor coefficients of the domain atendpoints of a bouncing ball orbit. From these one seeks to recover theTaylor coefficients and hence an analytic domain. So far the best resultis that one can recover a mirror symmetric analytic domain from itsspectrum. The second applicationis to geometry of eigenfunctions. Sogge and the proposer proved that the maximum possible growth rate of sup-norms of eigenfunctions onlyoccurs on boundarlylessRiemannian manifolds which contain recurrent points, such thata positive measure of geodesics leaving the point return in a fixedtime. One project is to prove this (if true) for bounded domains. A similar analysis is interesting for boundary values of eigenfunctions, whichare importantin boundary control theory. Hassell and the proposer have recentlyprovedthe ergodicity of boundary values of eigenfunctions when the billiardflowof the domain is ergodic, and this gave the first asymptotic estimatesofeven the L2 norms of the boundary values. Methods developed in thatpapershould have further applications to the geometry of eigenfunctions. Thethird project,joint with B. Shiffmanis to apply semiclassical methods to statistical algebraic geometry,i.e., tofinding statistical patterns in the zeros of polynomials. Among thepatterns foundso far are that zeros of systems of polynomials tend to repel indimension one,be liike a neutral gas in dimension two and attract in higherdimensions. Newproblems are to explore the dependence of the distribution of zeros onthe Newtonpolytope or on the number of monomials, as in Khovanski's fewnomialtheory.Semiclassical analysis is the area of mathematics which explores theclassicallimit of quantum mechancs, i.e.the limit where the fuzzy, jumpy smallscale behaviorof atoms and molecules merges with the solid, mechanical world weexperience.Planck's constant h measures the length scale. Although it arose inphysics,the same limit, as h tends to zero, arises in many problems inmathematics andscience, often quite disconnected from the original physicalbackground. Thisproposal is concerned with several problems of this kind. Several aretraditionaland well-known problems, for instance trying to determine a domain fromitsfrequencies of vibration. Using semiclassical analysis of a kind morefamiliar tophysicists than mathematicians (Balian-Bloch methods, Feynmandiagrams), the first project is to obtain the best results to date on the problem,can onehear the shape of a drum. The results so far indicate that this methodis betterthan any prior method. Another project is to view the degree of apolynomialas 1/h (inverse Planck constant) and to use semiclassical methods tofind newstatistical patterns in zeros and critical points of polynomials. Oneresult isthat the exponents which occur in the polynomials have a big impact onthepositions of the zeros, leading to a tunnelling theory of zerosanalogousto the tunnelling theory of electrons through barriers.