Problems in Spectral Theory and Analysis

这个项目的主要目标是开发分析工具来研究量子力学的主要对象薛定谔算符。该理论描述了一个物理系统，比如一群受到一定力的粒子，将如何随着时间的推移而变化。这对于预测电子、原子和分子等量子粒子的行为很重要。主要的研究人员将发展一些基本的方法来理解薛定谔算符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性质和输运。这些结果不仅很好地解释了物理和化学中的一些现象，而且在半导体等现代工程设备中也可能有丰硕的应用。本科生和研究生将有机会参与一些研究。这个项目旨在通过泛函、调和和几何分析的方法学习数学的几个方面。准周期薛定谔算符描述了在垂直于晶格面的磁通外加磁场作用下，二维晶层中电子的导电性。主要的研究人员将研究准周期算符的谱理论，包括谱跃迁，纯点谱区域的本征函数结构，以及奇异连续谱区域的谱测量的量子动力学。主要研究者还将研究非紧完备黎曼流形上的拉普拉斯谱理论。重点研究了渐近平坦流形和渐近双曲流形上嵌入拉普拉斯本质谱的特征值或奇异连续谱的存在性，其特征在于径向曲率。其目的是更好地理解几何量和本征解的性质之间的关系。最后，主要研究人员计划研究各类函数的时频平移的独立性，这被称为HRT猜想。首先证明了特殊构型的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.