现代调和分析是数学研究的核心学科之一，其不仅有丰富的理论知识，而且在复变函数、偏微风方程、解析数论以及数学物理等学科研究中有着广泛的应用。近年来，申请者致力于调和分析与微分算子交叉学科的重要论题的研究，取得了一系列重要的研究成果，并发表在CMP、IMRN、JFA等国际重要期刊上。在本项目中，申请者将继续应用调和分析方法对微分算子开展相关研究，其中包括薛定谔算子的色散估计及应用，热核估计与谱乘子理论等重要内容。在研究中，我们不仅需要调和分析的各种思想和方法，也需要泛函分析的谱理论和扰动技术。

Modern harmonic analysis is one of the core subjects of mathematics, which has not only very abundant theory, but also wide-ranging applications to several complex function,PDEs,analytic number theory and mathematical physics and so on. In recent years, the applicant has devoted to study important interacting topics related with harmonic analysis and differential operators, and achieved a series of innovative works published on several top international journals including CMP, IMRN and JFA and so on. In this project, the applicant will continue to study the topics related to differential operators by using methods from harmonic analysis, including the dispersive estimates of Schrödinger operator and its applications, as well as heat kernel estimates and spectral multiplier theory. In these studies, we need not only the ideas from harmonics analysis, but also the spectral theory and perturbation techniques from functional analysis.