Hamiltonian Systems and Related Phenomena

哈密​​顿系统和相关现象

• 批准号: 2000345
• 负责人: Wencai Liu
• 金额: $ 18万
• 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: 2020
• 资助国家: 美国
• 起止时间: 2020-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2000345&HistoricalAwards=false
• 关键词: Hamiltonian; Systems; Related; Phenomena

This project will study the dynamics of the Schrodinger operator that describes the wave function or state function of a quantum mechanical system. The principal investigator will develop fundamental tools to understand many types of phenomena in physics and chemistry, such as the quantum Hall effect, spin, crystals and quasicrystals. In particular, the project focuses on the conductance, transport, and localization-diffusion in both quasiperiodic and disordered media. The development of the rigorous theory is expected to have many applications, including to quantum information theory and semiconducting materials. The project provides research training opportunities for undergraduate and graduate students, and supports related outreach activities.This project aims at studying several topics in areas spanning mathematical physics, dynamical systems, harmonic analysis, geometric analysis, and algebraic geometry. The Principal Investigator will focus on transitions, hierarchical structures of eigenfunctions, quantum dynamics and spectral gaps for one dimensional quasiperiodic operators and further explore multifrequency and higher dimensional cases. The research will combine methods from algebraic geometry with tools from analysis to study the (ir)reducibility of Bloch, Fermi, and Floquet varieties arising from periodic elliptic operators. In the area of geometric analysis, the project emphasizes using piecewise constructions, gluing techniques, and the compact perturbation theory to investigate the transition of singular spectra and the generic phenomena of Laplacians on noncompact complete manifolds.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目将研究描述量子力学系统的波函数或状态函数的薛定谔算符的动力学。首席研究员将开发基本工具来理解物理和化学中的许多类型的现象，如量子霍尔效应，自旋，晶体和准晶体。该项目特别关注准周期和无序介质中的电导、输运和定位扩散。这一严谨理论的发展有望在量子信息理论和半导体材料等领域得到广泛应用。该项目为本科生和研究生提供研究培训机会，并支持相关的外展活动。该项目旨在研究数学物理、动力系统、谐波分析、几何分析和代数几何等领域的几个主题。首席研究员将重点研究一维准周期算子的跃迁、特征函数的层次结构、量子动力学和谱间隙，并进一步探索多频和高维情况。该研究将结合代数几何的方法和分析的工具来研究由周期椭圆算子引起的Bloch， Fermi和Floquet变体的（ir）可约性。在几何分析方面，本项目强调使用分段构造、胶合技术和紧致摄动理论来研究奇异谱的跃迁和拉普拉斯算子在非紧致完全流形上的一般现象。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Irreducibility of the Bloch variety for finite-range Schrödinger operators

有限范围薛定谔算子的布洛赫簇的不可约性

• DOI: 10.1016/j.jfa.2022.109670
• 发表时间: 2022
• 期刊: Journal of Functional Analysis
• 影响因子: 1.7
• 作者: Fillman, Jake;Liu, Wencai;Matos, Rodrigo
• 通讯作者: Matos, Rodrigo

Spacetime quasiperiodic solutions to a nonlinear Schrödinger equation on Z

Z 上非线性薛定谔方程的时空准周期解

• DOI: 10.1063/5.0166183
• 发表时间: 2024
• 期刊: Journal of Mathematical Physics
• 影响因子: 1.3
• 作者: Kachkovskiy, Ilya;Liu, Wencai;Wang, Wei-Min
• 通讯作者: Wang, Wei-Min

Topics on Fermi varieties of discrete periodic Schrödinger operators

关于离散周期薛定谔算子的费米簇的主题

• DOI: 10.1063/5.0078287
• 发表时间: 2022
• 期刊: Journal of Mathematical Physics
• 影响因子: 1.3
• 作者: Liu, Wencai

Power law logarithmic bounds of moments for long range operators in arbitrary dimension

• DOI: 10.1063/5.0138325
• 发表时间: 2022-12
• 期刊: Journal of Mathematical Physics
• 影响因子: 1.3
• 作者: Wencai Liu

Fermi Isospectrality of Discrete Periodic Schrödinger Operators with Separable Potentials on $$\mathbb {Z}^2$$

$$mathbb {Z}^2$$ 上具有可分离势的离散周期薛定谔算子的费米同谱性

• DOI: 10.1007/s00220-022-04575-8
• 发表时间: 2023
• 期刊: Communications in Mathematical Physics
• 影响因子: 2.4
• 作者: Liu, Wencai