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ON THE GENERALIZED FOURIER TRANSFORMS ASSOCIATED RELATIVISTIC SCHRODINGER OPERATORS

关于广义傅立叶变换相关相对论薛定谔算子

基本信息

批准号：
09640212
负责人：
UMEDA Tomio
金额：
$ 1.92万
依托单位：
HIMEJI INSTITUTE OF TECHNOLOGY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 2000
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640212/
关键词：
SPECTRAL THEORY SCATTERING THEORY RELATIVISTIC SCHRODINGER OPERATORS ELGENFUNCTION EXPANSIONS

项目摘要

This project is an attempt to make an approach to spectral and scattering theory for relativistic Schrodinger operators. The aim of the project is to investigate the generalized Fourier transforms through analyzing the generalized eigenfunctions in great detail. Below is what has been shown in this project.1. Completeness of the generalized eigenfunctions (the massive case)It is shown that the family of generalized eigenfunctions of the relativistic Schrodiger operator is complete in the subspace of continuity.2. Completeness of the generalized eigenfunctions (the massless case)The same fact as in Result 1 is shown. The point is a successful treatment of the difficulties which are specific to this case.3. A characterization of the generalized eigenfunctions (the massless and 3-dimensional case)Based on an explicit computation of the resolvent kernel of the square-root of the minus Laplacian, the generalized eigenfunctions are characterized as the unique solutions to the Lippmaun-Schwinger type integral equations4. The action of the square-root of the minus Laplacian on distributionsSharp estimates on the square-root of minus Laplacian in weighted Sobolev spaces and the radiation conditions are derived.In connection with Result 3, we have recognized that it is possible to make detailed analysis on the regularity of the generalized eigenfunctions as well as on the difference of the generalized eigenfunctions from the plane wave solutions. With this respect, we still continue the research. Result 4 was not contained in our initial plan, although it is closey related with the aim of our project. It seems, however, to bear an important aspect of mathematics which has been ignored so far. For this reason, we continue making research on the action of the square-root of the minus Laplacian, and shall try to extend the result.

本项目是对相对论性薛定谔算子的光谱和散射理论进行探讨的一种尝试。本课题的目的是通过对广义特征函数的详细分析来研究广义傅里叶变换。下面是在这个项目中所显示的。广义本征函数的完备性（质量情况）证明了相对论性Schrodiger算子的广义本征函数族在连续子空间上是完备的。广义本征函数的完备性（无质量情况下）证明了与结果1相同的事实。关键是成功地处理了这个案例所特有的困难。广义本征函数的表征（无质量和三维情况）基于负拉普拉斯平方根的解核的显式计算，将广义本征函数表征为Lippmaun-Schwinger型积分方程的唯一解4。给出了负拉普拉斯算子的平方根对分布的作用，以及加权Sobolev空间中负拉普拉斯算子的平方根的夏普估计和辐射条件。结合结果3，我们认识到可以详细分析广义本征函数的规律性以及广义本征函数与平面波解的差异。在这方面，我们仍在继续研究。结果4并没有包含在我们最初的计划中，虽然它与我们项目的目的密切相关。然而，它似乎包含了迄今为止一直被忽视的数学的一个重要方面。因此，我们将继续研究负拉普拉斯式的平方根的作用，并尝试推广这一结果。