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

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.