Harmonic analysis techniques in spectral theory

围绕“傅立叶限制猜想”的一系列问题一直是现代调和分析发展的推动力。近年来，在偏微分方程、组合学、几何分析和数论方面取得了令人振奋的进展，取得了惊人的应用。然而，谱理论中的大多数应用仍然基于经典的Stein-Tomas定理。目前尚不清楚调和分析的最新进展(如解耦、多项式分割、多线性方法和尺度上的Bourain-Guth归纳)是否以及如何应用于谱理论。该研究计划旨在通过开发一种新的方法来解决这个问题，该方法将现代调和分析技术与经典泛函分析和复变分析相结合。这些技术有望在涉及微分算子的各种频谱问题中得到应用。重点放在具有复势的薛定谔算子上，这是一个在当代很有意义的课题。处理这些算子的技术是有限的，这主要是由于缺乏强大的谱定理和变分方法。事实证明，调和分析工具在这方面非常有成效，并在解决该领域的重大公开问题方面发挥了重要作用。从广义上讲，人们想要了解，具有实值位势的薛定谔算子的经典理论有多少仍然适用于复值位势；在许多情况下，答案是“不是很多”。复势自然地出现在许多不同的问题中，如偏微分方程组的局部可解性、Kramers-Fokker-Planck方程、阻尼波方程、共振研究以及非线性偏微分方程组的稳定性分析。它们的光谱特性可能非常狂野和不直观，但它们仍然远未被完全理解。

The circle of problems around the "Fourier restriction conjecture" has been a driving force behind the development of modern harmonic analysis. Recent years have seen exciting advances that have lead to spectacular applications in PDE, combinatorics, geometric analysis and number theory. However, most applications in spectral theory are still based on the classical Stein-Tomas theorem. It is currently not understood if and how the state-of-the-art advances in harmonic analysis (such as decoupling, polynomial partitioning, multilinear methods and the Bourgain--Guth induction on scales) can be applied to spectral theory. The research programme aims to address this issue by developing a new approach that combines modern harmonic analysis techniques with classical functional and complex analysis.These techniques are expected to find applications in a variety of spectral problems involving differential operators. The focus is on Schrödinger operators with complex potentials, a subject of considerable contemparary interest. The techniques to deal with these operators are limited, mainly due to a lack of the powerful spectral theorem and variational methods. Harmonic analysis tools have turned out to be extremely fruitful in this regard and have been instrumental in solving major open problems in the field. In a broad sense, one would like to understand how much of the classical theory of Schrödinger operators with real-valued potentials survives for complex-valued potentials; the answer in many cases is "not very much". Complex potentials appear naturally in many different problems, for instance, local solvability of PDE, the Kramers-Fokker-Planck equation, the damped wave equation, the study of resonances and in the stability analysis of nonlinear PDE. Their spectral properties can be very wild and unintuitive, and they are still far from being completely understood.