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

小分母问题和准周期运动自然出现在具有多个不相称周期运动频率的经典系统和量子系统中。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）、固体物理（准晶体）、数学物理（准周期薛定谔算子，或者更一般地说，具有局域化的系统中的时间依赖动力学）和偏微分方程（具有周期系数的非线性薛定谔方程和波动方程）中。分析这类问题需要处理小分母；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。在准周期算子的情况下，KAM方法的主要局限性是由于共振结构复杂，很难应用于真正的多维系统。替代方法（基于格林函数估计的方法）没有这些维度限制。直到最近，这些方法还没有像KAM那样在参数去除方向上灵活。然而，由于该项目的主要研究者（pi）最近的工作，这种情况目前正在发生变化。该项目包括研究和培训活动，以发展和完善这些新方法，并将它们应用于涉及拟周期薛定谔算子和非线性偏微分方程的问题的研究，获得以前无法获得的多维和算术结果。这些在上述所有领域都有潜在的应用。该建议的技术核心是格拟周期算子的格林函数估计的非摄动方法的发展，假设频率参数被限制在环面的子流形上。这些问题自然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（NLW）方程的分析中，直到布尔甘-卡赫科夫斯基的工作才得以实现，然而，这只是第一步，因为它严重依赖于二维环境。这些方法将被应用于构造NLS和NLW方程的新类时空准周期解，通过解除当前克雷格-韦恩-布尔格恩方法的维度和算术限制。这些方法也有望用于构建全维的KAM tori。来自多个方向的pi的最新进展也允许，第一次考虑多维准周期算子的算法定位结果，这是受到Jitomirskaya和Liu最近获得的尖锐结果的启发。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。

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.