Problems in Spectral Theory and Analysis

该项目的主要目标是开发分析工具来研究薛定谔算子，量子力学的主要对象。该理论描述了一个物理系统，比如一堆受到某种力作用的粒子，将如何随着时间的推移而变化。这对于预测量子粒子，如电子，原子和分子的行为很重要。主要研究者将发展一些基本的方法来理解薛定谔算子的光谱和量子动力学行为。特别是，该项目侧重于准周期介质的电导特性和传输。这些结果不仅对物理和化学中的一些现象提供了很好的解释，而且可能在半导体等现代工程器件中有富有成效的应用。本科生和研究生将有机会参与部分研究。本项目旨在通过泛函分析、调和分析和几何分析的方法研究数学的几个方面。准周期薛定谔算符描述了二维晶体层中电子的电导率，该二维晶体层受到垂直于晶格平面作用的通量的外部磁场。主要研究者将研究准周期算子的谱理论，包括谱跃迁，纯点谱区本征函数的结构，以及奇异连续谱区谱测度的量子动力学。主要研究者还将研究非紧完备黎曼流形上拉普拉斯算子的谱理论。 重点是存在的特征值或奇异连续谱嵌入到本质谱的拉普拉斯算子的渐近平坦和渐近双曲流形，其特征在于径向曲率。目的是更好地理解几何量和本征解性质之间的关系。最后，主要研究者计划研究各类函数的时频平移的独立性，即HRT猜想。 重点是证明特殊构型的HRT猜想和指数衰减函数的HRT猜想。

The primary goal of this project is to develop analytic tools to study Schrodinger operators, the main object of quantum mechanics. The theory describes how a physical system, say a bunch of particles subject to certain forces, will change over time. This is important to predict the behavior of the quantum particles, such as electrons, atoms and molecules. The principal investigator will develop some fundamental methods to understand the spectra and quantum dynamical behavior of Schrodinger operators. In particular, the project focuses on the conductance properties and transport of quasi-periodic media. The results not only provide a good explanation for some phenomena in physics and chemistry, but may also have fruitful applications to modern engineering devices such as semiconductors. Undergraduate and graduate students will have opportunities to participate in some of the research.This project aims at studying several aspects in mathematics by methods of functional, harmonic and geometric analysis. Quasi-periodic Schrodinger operators describe the conductivity of electrons in a two-dimensional crystal layer subject to an external magnetic field of flux acting perpendicular to the lattice plane. The principal investigator will investigate the spectral theory of quasi-periodic operators, including spectral transitions, structure of eigenfunctions in the pure point spectrum regime, and quantum dynamics of spectral measures in the singular continuous spectrum regime. The principal investigator will also study the spectral theory of Laplacians on noncompact complete Riemannian manifolds. The focus is on the existence of eigenvalues or singular continuous spectra embedded into essential spectra of Laplacians on asymptotically flat and on asymptotically hyperbolic manifolds, as characterized by the radial curvature. The goal is to better understand relations between geometric quantities and properties of eigensolutions. Lastly the principal investigator plan to study the independence of the time-frequency translates of various classes of functions, which is stated as HRT conjecture. The priority is to prove the HRT conjecture for special configurations and HRT conjecture for exponentially decaying functions.