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经典和量子系统中，这些系统具有多个不可公度的周期运动频率。这类系统的例子存在于天体力学（行星轨道）、生物学（人口动力学）、固态物理学（准晶体）、数学物理学（准周期薛定谔算子，或者更一般地说，局部化系统中的时间相关动力学）和偏微分方程（非线性薛定谔方程和周期系数的波动方程）。对这些问题的分析需要处理一些小的干扰因素;换句话说，要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，这些问题已经接近Kolmogorov-Arnold-Moser（KAM）型技术。在准周期算子的设置下，KAM方法的主要局限性在于，由于共振结构的复杂性，它们很难应用于真正的多维系统。替代方法（基于绿色函数估计的方法）没有这些尺寸限制。直到最近，这些方法还没有像KAM那样灵活地去除参数。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。该项目涉及研究和培训活动，以发展和完善这些新方法，并将其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。这些都有潜在的应用在上述所有领域的建议的技术核心是发展的非微扰方法的绿色的功能估计格准周期运营商，假设频率参数被限制到一个环面的子流形。这样的问题自然出现在多粒子准周期算子以及非线性薛定谔（NLS）和非线性波（NLW）方程的分析中，并且直到Bourgain-Kachkovskiy的工作才被访问，然而，这只是第一步，因为它显著依赖于二维设置。这些方法将被应用到构建新的类的时空拟周期解的NLS和NLW方程，通过解除目前的维度和算术的Craig-Wayne-Bourgain方法的限制。预计这些方法将允许构建全维KAM环面。最近的进步，从多个方向的PI也允许，第一次，考虑多维准周期运营商的算术本地化结果，最近获得的Jitomirskaya和Liu.This奖反映了NSF的法定使命，并已被认为是值得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的支持。

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.