Global harmonic analysis and quantum dynamics

AbstractAward：DMS-1206527首席研究员：Steve Zelditch本提案中的项目涉及全局调和分析和渐近几何，重点是黎曼流形上拉普拉斯算子的本征值和本征函数，以及复几何中的Bergman核和Bergman度量。 在黎曼背景下，主要问题涉及拉普拉斯算子的本征值和本征函数。 一个重点是量子遍历本征函数的节点和临界点集。C. Sogge和主要研究者最近发现了一个恒等式，将节点集积分与本征函数的L1范数联系起来，该恒等式用于给出节点集体积的下界。PI已经证明了临界点的类似恒等式，并计划使用它来研究临界点的分布。 PI还计划继续研究Grauert管本征函数的解析延拓，作为研究复零点和临界点的工具。利用J. Toth关于超曲面的量子遍历限制的新结果，PI计划计算测地线和节点集的交叉点的分布。 在Kaehler几何中，本计画将发展Kaehler度量在一固定类中被Bergman度量所逼近的理论。特别是PI正在研究Kaehler度量空间中测地线的初始值问题。鲁宾斯坦我们已经表明，在大多数方向上，长期以来解决这个问题都存在障碍。另一个项目是关于“随机度规”，即Kaehler度规空间上的概率测度，它是通过Bergman度规的对称空间来近似定义的。近100年来，量子力学和量子场论是物理学的基础理论。量子物理学不是一种直观的理论，不能像力学、电学和磁学那样与普通世界联系起来。 因此，研究具有基本量子特征的简单模型具有重要意义。自1800年Chladni以来，人们一直在研究振动鼓和表面，但Chladni提出的基本问题是完全开放的。如果一个人把沙子倒在振动鼓上，它会沉淀成“节点模式”，即振动鼓不上下移动而是保持静止的点。 这些节点模式使我们能够可视化鼓的振动模式。随着振动频率的增加，图案变得越来越复杂。youtube上有把沙子倒在振动鼓上的视频，人们可以看到这些图案，但很难预测它们会是什么样子。物理学家推测，如果在鼓上打台球是“混乱的”（即不可预测的，本质上是随机的），那么节点线也会像随机曲线。 首席研究员已经表明，如果将振动模式扩展到复域（即复数世界），那么物理学假设是正确的。 PI现在正试图证明沿着台球的沿着轨迹也是一样的。人们还希望了解“临界点”，即，鼓振动到其顶部和底部位置的点。 同样的技术也适用于更深奥的量子引力物理学，首席研究员正在与两位物理学家合作。

AbstractAward: DMS-1206527 Principal Investigator: Steve ZelditchThis projects in this proposal concern global harmonic analysis and asymptotic geometry, with emphases on eigenvalues and eigenfunctions of the Laplacian on Riemannian manifolds, and on Bergman kernels and Bergman metrics in complex geometry. In the Riemannian setting, the main problems concern eigenvalues and eigenfunctions of the Laplacian. One focus is on nodal and critical point sets of quantum ergodic eigenfunctions. C. Sogge and the principal investigator recently found an identity relating nodal set integrals to L1 norms of eigenfunctions, which was used to give a lower bound on volumes of nodal sets. The PI has proved an analogous identity for critical points and plan to use it to study the distribution of critical points. The PI also plans to continue a study of the analytic continuation of eigenfunctions to Grauert tubes as a tool to study complex zeros and critical points. Using a new result with J. Toth on quantum ergodic restriction to hypersurfaces, the PI plans to work out the distribution of intersections of geodesics and nodal sets. In Kaehler geometry, this project will develop the approximation theory of Kaehler metrics in a fixed class by Bergman metrics. In particular the PI is studying the initial value problem for geodesics in the space of Kaehler metrics in joint work with Y. Rubinstein. We have shown that there are obstructions to solving it for long times in most directions. Another project concerns "random metrics," i.e. probability measures on the space of Kaehler metrics which are defined by approximating it by the symmetric space of Bergman metrics.For almost 100 years now, quantum mechanics and quantum field theory are the basic theories in physics. Quantum physics is not an intuitive theory that can be related to the ordinary world in the way that mechanics or electricity and magnetism can be. So it is important to study simple models which exhibit the basic quantum features. Vibrating drums and surfaces have been studied since Chladni in 1800 and yet the basic questions asked by Chladni are completely open. If one pours sand on a vibrating drum, it will settle into "nodal patterns" which are the points where the vibrating drum is not moving up and down but staying still. These nodal patterns allow us to visualize the modes of vibration of the drum. As the frequency of vibration increases, the pattern becomes more and more complicated. There exist youtube videos of sand poured on vibrating drums where one can see the patterns, but it is very hard to predict what they will look like. Physicists conjecture that if the billiards played on the drum are "chaotic" (i.e unpredictable and essentially random), then the nodal lines will also be like random curves. The principal investigator has shown that if one extends the modes of vibration into the complex domain (i.e. the world of complex numbers) then the physics conjectures are true. The PI is now trying to prove the same along trajectories of the billiard balls. One also hopes to understand the "critical points," i.e., the points where the drum vibrates to its top and bottom positions. The same techniques apply to the much more abstruse physics of quantum gravity, and the principal investigator is collaborating with two physicists on that.