(Semi)algebraic Geometry in Schrödinger Operators and Nonlinear Hamiltonian Partial Differential Equations

薛定谔算子和非线性哈密顿偏微分方程中的（半）代数几何

• 批准号: 2246031
• 负责人: Wencai Liu
• 金额: $ 27.09万
• 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: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2026-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2246031&HistoricalAwards=false
• 关键词: Semi; algebraic; Geometry; Schr; dinger

The Schrödinger equation is fundamental in quantum mechanics as it describes the behavior of particles, such as electrons, in a physical system. This project aims to develop mathematical tools to investigate various phenomena observed in solid-state physics, condensed matter physics, and optics. The mathematical understanding of these phenomena could lead to numerous applications, such as the design of quantum computing and quantum communication devices, and the development of novel semiconducting materials. Research training opportunities will be provided for both undergraduate and graduate students.The objective is to explore both linear and nonlinear Hamiltonian systems through various mathematical tools such as (semi)algebraic geometry, mathematical physics, and dynamical systems. The project will focus on three main areas. The first area will center on analyzing spectral transitions, the hierarchical structures of eigenfunctions, quantum dynamics, and spectral gaps of quasiperiodic operators. The second area will involve combining methods from algebraic geometry with analysis tools to study the irreducibility of Bloch and Fermi varieties, and the inverse spectral problems of periodic graph operators. Finally, the project will develop new techniques from semi-algebraic geometry and perturbation theory to study the quasi-periodic and almost periodic in time solutions of nonlinear Schrödinger and wave equations.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.

薛定谔方程是量子力学的基础，因为它描述了物理系统中粒子（例如电子）的行为。该项目旨在开发数学工具来研究在固态物理学、凝聚态物理学和光学中观察到的各种现象。对这些现象的数学理解可能会带来许多应用，例如量子计算和量子通信设备的设计以及新型半导体材料的开发。将为本科生和研究生提供研究培训机会。目标是通过（半）代数几何、数学物理和动力系统等各种数学工具探索线性和非线性哈密顿系统。该项目将重点关注三个主要领域。第一个领域将集中于分析谱跃迁、本征函数的层次结构、量子动力学和准周期算子的谱间隙。第二个领域将涉及将代数几何方法与分析工具相结合，研究布洛赫和费米簇的不可约性，以及周期图算子的逆谱问题。最后，该项目将开发半代数几何和微扰理论的新技术，以研究非线性薛定谔和波动方程的准周期和近周期时间解。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力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

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

Fermi isospectrality for discrete periodic Schrödinger operators

离散周期薛定谔算子的费米同谱

• DOI: 10.1002/cpa.22161
• 发表时间: 2023
• 期刊: Communications on Pure and Applied Mathematics
• 影响因子: 3
• 作者: Liu, Wencai

Algebraic properties of the Fermi variety for periodic graph operators

周期图算子费米簇的代数性质

• DOI: 10.1016/j.jfa.2023.110286
• 发表时间: 2024
• 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

Floquet isospectrality for periodic graph operators

周期图算子的 Floquet 同谱性

• DOI: 10.1016/j.jde.2023.08.009
• 发表时间: 2023
• 期刊: Journal of Differential Equations
• 影响因子: 2.4
• 作者: Liu, Wencai