Problems in Spectral Theory and Analysis

谱理论与分析中的问题

• 批准号: 2015683
• 负责人: Wencai Liu
• 金额: $ 3.06万
• 依托单位: Texas A&M University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-09-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2015683&HistoricalAwards=false
• 关键词: Problems; Spectral; Theory; Analysis

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.

该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——薛定谔算子。该理论描述了一个物理系统，比如一堆受某种力影响的粒子，将如何随着时间而变化。这对于预测量子粒子（如电子、原子和分子）的行为非常重要。首席研究员将开发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。该项目特别关注准周期介质的电导性质和输运。这一结果不仅对一些物理和化学现象提供了很好的解释，而且在半导体等现代工程器件上也具有丰富的应用价值。本科生和研究生将有机会参与其中的一些研究。本项目旨在用泛函、调和和几何分析的方法研究数学中的几个方面。准周期薛定谔算符描述了在垂直于晶格平面的外磁场作用下二维晶体层中电子的导电性。主要研究拟周期算符的谱理论，包括谱跃迁、纯点谱域本征函数的结构以及奇异连续谱域谱测度的量子动力学。首席研究员还将研究非紧化完全黎曼流形上拉普拉斯算子的谱理论。重点讨论了渐近平面和渐近双曲流形上拉普拉斯算子本质谱中特征值或奇异连续谱的存在性。目的是更好地理解几何量和特征解的性质之间的关系。最后，主要研究者计划研究各类函数时频转换的独立性，并将其表述为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。

项目成果

Anderson localization for multi-frequency quasi-periodic operators on $$\pmb {\mathbb {Z}}^{d}$$

• DOI: 10.1007/s00039-020-00530-8
• 发表时间: 2019-08
• 期刊: Geometric and Functional Analysis
• 影响因子: 2.2
• 作者: S. Jitomirskaya;Wencai Liu;Yunfeng Shi

Revisiting the Christ–Kiselev’s multi-linear operator technique and its applications to Schrödinger operators

重温 Christ—Kiselev—的多线性算子技术及其在薛定谔算子中的应用

• DOI: 10.1088/1361-6544/abbd85
• 发表时间: 2021
• 期刊: Nonlinearity
• 影响因子: 1.7
• 作者: Liu, Wencai

Sharp spectral transition for eigenvalues embedded into the spectral bands of perturbed periodic operators

嵌入扰动周期算子谱带中的特征值的急剧谱转变

• DOI: 10.1007/s11854-020-0111-x
• 发表时间: 2020
• 期刊: Journal d'Analyse Mathématique
• 作者: Liu, Wencai;Ong, Darren C.
• 通讯作者: Ong, Darren C.

Exact dynamical decay rate for the almost Mathieu operator

• DOI: 10.4310/mrl.2020.v27.n3.a8
• 发表时间: 2018-12
• 期刊: Mathematical Research Letters
• 影响因子: 1
• 作者: S. Jitomirskaya;Helge Kruger;Wencai Liu

Ambarzumian-type Problems for Discrete Schrödinger Operators

离散薛定谔算子的 Ambarzumian 型问题

• DOI: 10.1007/s11785-021-01169-5
• 发表时间: 2021
• 期刊: Complex Analysis and Operator Theory
• 影响因子: 0.8
• 作者: Hatinoğlu, Burak;Eakins, Jerik;Frendreiss, William;Lamb, Lucille;Manage, Sithija;Puente, Alejandra
• 通讯作者: Puente, Alejandra