Pseudodifferential Operators and Schrodinger Equations

本研究项目的目的是研究伪微分算子在薛定谔方程理论中的应用。在这方面已经有很多研究工作。本课题主要利用伪微分算子理论研究薛定谔算子的谱理论。在伪微分算子理论中，我们得到了伪微分算子的有界性定理和紧性定理，并得到了Garding不等式尖锐形式的结果。我们的项目有很多研究者，我们处理的课题非常广泛，所以我们的成员参加了很多关于偏微分方程、伪微分算子和数学物理的会议和研讨会。此外，我们还邀请了来自波茨坦的E.Schrohe教授，来自乔治亚大学的C.Heil教授。来自叶纳的h .特里贝尔教授等等。他们拜访了我们，回顾了我们的研究，并做了几次有趣的演讲，他们对我们的研究非常有用。我们得到了由具有磁势的量子化哈密顿量产生的伪微分算子谱的不变性的一些结果，这些结果与Schrohe教授的工作密切相关。此外，我们还得到了多维多小波的一些结果，这些结果是由伪微分算子的维格纳分布表示而来的。

The purpose of the present research project is to investigate the application of pseudodifferential operators to the theory of Schrrodinger equations. There has already been many research works in this subject. In this project we investigate mainly the spectral theory of Schrodinger operators by using the theory of pseudodifferential operators. In the theory of pseudodifferential operators we obtained boundedness theorems and compactness of pseudodifferential operators and also we got the results for sharp form of Garding's inequality.Our project has many investigators and the subject which we treated is very wide, so our member attended many meetings and workshop about partial differential equations, pseudodifferential operators and mathematical phisics.Moreover we invited Professor E.Schrohe from Potzdam, Professor C.Heil from Georgia Thec., Professor H.Triebel from Yena and so on. They visited us for reviewing our research and gave several interesting talks and they were very useful for our research. We get some results on the invariance of spectral of the pseudodifferential operators which arise from the quantized Hamiltonian with magnetic potentials, which are closely related to the work of Professor Schrohe. Moreover we get some results on the muti-dimensional multi-wavelet, the result which comes from the representations of pseudidifferen-tial operators by using the Wigner distributions.