调和分析在薛定谔算子色散估计与谱研究中的应用

Modern harmonic analysis is one of the core mathematics research fields, which is deeply related to many other important subjects such as several complex variables, partial differential equations, mathematics and physics and so on. By using the methods of harmonic analysis, the applicant has achieved a series of innovative works in such topics as dispersive equations and the spectra and semigroup of Schrodinger operators, which have been published at some top international Journals " J. Funct. Anal." and " Com. PDEs " and so on. In this project, the applicants will continue to study the interactions between Harmonic analysis and Mathematics with Phyisics. Specially, this project will includes the dispersive estimates of Schrodinger type group with potentials, the Lp-Lq uniform estimates of resolvent and spectral analysis. In the specific investigations, one needs to widely use the methods and tools from harmonic analysis, spectral theory and number theory. For instance, the oscillatory integrals theory and interpolation theorems are used to study disperse estimates of the operator group, as well as the spectral asymptotic distribution and the count methods of lattice points are also involved to deal with the problems on manifolds. The expected results can be applied to some important topics such as the well-posedness of nonlinear dispersive equation, spectral multipliers and also to the unique continuation problem.

现代调和分析一直是数学研究的核心领域之一，与多复变分析、偏微分算子、数学物理以及对称空间等学科领域有着十分广泛密切的联系；利用调和分析方法，近年来申请者已在色散方程、薛定谔算子半群与谱等论题取得了一些好的创新成果，并在 "J. Funct. Anal."，"Comm. PDEs"等多个国际数学刊物上发表。 本项目申请者将继续研究调和分析与数学物理交叉领域若干重要主题，其中包括带位势的薛定谔算子群的色散估计，预解式算子的一致Lp-Lq 估计与谱分析等内容。在项目具体研究中，我们需要广泛使用调和分析方法、谱理论以及数论等重要工具，如需要振荡积分理论和插值定理来研究薛定谔型方程的色散估计，也需要充分利用谱Weyl 分布渐进结果及区域上格点计数数论方法来研究流形上问题等。所得结果将能应用于非线性色散方程的适定性、谱乘子有界性及唯一连续性等论题。

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,而且有着广泛的应用。近四年来，主持人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，已发表学术论文13篇(其中12篇SCIE/SSCI），发表杂志主要包括Comm .Math .Phys., Int.Math. Res.Not.(IMRN), J.Funct.Anal.等国际重要数学期刊。主要业绩有:(1)与J.Bourgain教授等合作，在紧流形上得到了Laplace-Beltrami算子最佳的一致Sobolev估计，回答了C.Kenig等人公开提出的问题;(2)建立了高阶微分算子的热核估计并研究与之相关的函数空间与奇异积分算子理论，推广了经典调和分析若干重要结果;(3)深入研究了薛定谔方程解的Lp-Lq估计和Strichartz估计，并应用于非线性薛定谔方程解的适定性和散射问题研究。