FRG: Collaborative Research: Non-Perturbative Analysis for Multi-Dimensional Quasiperiodic Systems

FRG：协作研究：多维准周期系统的非微扰分析

• 批准号: 2052572
• 负责人: Wencai Liu
• 金额: $ 41.4万
• 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: 2021
• 资助国家: 美国
• 起止时间: 2021-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2052572&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Non; Perturbative

Small denominator problems and quasiperiodic motion appear naturally in classical and quantum systems that have multiple incommensurate frequencies of periodic motion. Examples of such systems exist in celestial mechanics (planetary orbits), biology (population dynamics), solid state physics (quasicrystals), mathematical physics (quasiperiodic Schrodinger operators, or, more generally, time-dependent dynamics in systems with localization), and partial differential equations (non-linear Schrodinger and wave equations with periodic coefficients). The analysis of such problems requires dealing with small denominators; in other words, understanding how often and in what pattern would the system return to a state that is very close to the initial state. Traditionally, these problems have been approached by Kolmogorov-Arnold-Moser (KAM)-type techniques. In the setting of quasiperiodic operators, the main limitations of KAM methods is that they are very difficult to apply to truly multi-dimensional systems, due to the complicated structure of resonances. Alternative approaches (methods based on estimates of Green's functions) do not have these dimensional restrictions. Until recently, those methods have not been as flexible as KAM in the direction of parameter removal. However, this is currently changing largely due to the recent works of the principal investigators (PIs) of this project. The project involves research and training activities towards developing and refining these new methods and applying them to the study of problems involving quasiperiodic Schrodinger operators and nonlinear partial differential equations, obtaining previously inaccessible multi-dimensional and arithmetic results. These have potential applications in all the fields mentioned above.The technical heart of the proposal is the development of non-perturbative methods for Green’s function estimates for lattice quasiperiodic operators, assuming that the frequency parameter is restricted to a submanifold of a torus. Such problems appear naturally in the analysis of multi-particle quasiperiodic operators as well as nonlinear Schrodinger (NLS) and nonlinear wave (NLW) equations, and have been inaccessible until the work of Bourgain–Kachkovskiy which, however, is only the first step since it significantly relies on the two-dimensional setting. These methods will be applied to constructing new classes of spacetime quasiperiodic solutions of the NLS and NLW equations, by lifting the current dimensional and arithmetic restrictions of the Craig–Wayne–Bourgain approach. It is also expected that these methods will allow to construct full-dimensional KAM tori. Recent advances by the PIs from multiple directions also allow, for the first time, to consider arithmetic localization results for multi-dimensional quasiperiodic operators, motivated by recent sharp results obtained by Jitomirskaya and Liu.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.

在具有多个无公度周期运动频率的经典和量子系统中，自然会出现小分母问题和准周期运动。这种系统的例子存在于天体力学(行星轨道)、生物学(种群动力学)、固态物理(准晶)、数学物理(准周期薛定谔算子，或者更一般地，具有局域化的系统中的依赖时间的动力学)和偏微分方程组(具有周期系数的非线性薛定谔方程和波动方程)。对这类问题的分析需要处理小分母；换句话说，了解系统返回到非常接近初始状态的状态的频率和模式。传统上，这些问题是通过Kolmogorov-Arnold-Moser(KAM)型技术来解决的。在拟周期算符的设置下，KAM方法的主要局限性是由于共振的复杂结构，它们很难应用于真正的多维系统。其他方法(基于格林函数估计的方法)没有这些维度限制。直到最近，这些方法在参数删除方向上还没有KAM那么灵活。然而，这种情况目前正在发生变化，这主要是由于该项目的主要调查人员(PI)最近的工作。该项目包括研究和培训活动，以开发和完善这些新的方法，并将它们应用于研究涉及准周期薛定谔算子和非线性偏微分方程的问题，获得以前无法获得的多维和算术结果。该方案的技术核心是发展格点拟周期算子格林函数估计的非摄动方法，假设频率参数被限制在环面的一个子流形上。这些问题自然地出现在多粒子准周期算符以及非线性薛定谔(NLS)和非线性波动(NLW)方程的分析中，直到Bourain-Kachkovski的工作才能解决，然而，这只是第一步，因为它严重依赖于二维设置。通过解除Craig-Wayne-Bourain方法的维度和算术限制，这些方法将被应用于构造NLS和NLW方程的新的时空拟周期解。人们还期望这些方法能够构建全维的Kam Tori。PIs最近从多个方向取得的进展也首次允许考虑多维准周期算子的算术本地化结果，这是由Jitomirskaya和Liu最近获得的尖锐结果所推动的。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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

Irreducibility of the Fermi variety for discrete periodic Schrödinger operators and embedded eigenvalues

离散周期薛定谔算子和嵌入特征值的费米簇的不可约性

• DOI: 10.1007/s00039-021-00587-z
• 发表时间: 2022
• 期刊: Geometric and Functional Analysis
• 影响因子: 2.2
• 作者: Liu, Wencai

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

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

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