Project/Area Number  09640212 
Research Category 
GrantinAid for Scientific Research (C).

Section  一般 
Research Field 
解析学

Research Institution  HIMEJI INSTITUTE OF TECHNOLOGY 
Principal Investigator 
UMEDA Tomio SCIENCE, HIMEJI INSTITUTE OF TECHNOLOGY PROFESSOR, 理学部, 教授 (20160319)

CoInvestigator(Kenkyūbuntansha) 
HOSHIRO Toshihiko SCIENCE, HIMEJI INSTITUTE OF TECHNOLOGY ASSOCIATE PROFESSOR, 理学部, 助教授 (40211544)
IWASAKI Chisato SCIENCE, HIMEJI INSTITUTE OF TECHNOLOGY PROFESSOR, 理学部, 教授 (30028261)
平野 克博 姫路工業大学, 理学部, 講師 (90316034)

Project Fiscal Year 
1997 – 2000

Project Status 
Completed(Fiscal Year 2000)

Budget Amount *help 
¥3,000,000 (Direct Cost : ¥3,000,000)
Fiscal Year 2000 : ¥900,000 (Direct Cost : ¥900,000)
Fiscal Year 1999 : ¥600,000 (Direct Cost : ¥600,000)
Fiscal Year 1998 : ¥800,000 (Direct Cost : ¥800,000)
Fiscal Year 1997 : ¥700,000 (Direct Cost : ¥700,000)

Keywords  SPECTRAL THEORY / SCATTERING THEORY / RELATIVISTIC SCHRODINGER OPERATORS / ELGENFUNCTION EXPANSIONS / スペクトル理論 / 散乱理論 / 相対論的シュレディンガー作用素 / 固有関数展開 
Research Abstract 
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 3dimensional case) Based on an explicit computation of the resolvent kernel of the squareroot of the minus Laplacian, the generalized eigenfunctions are characterized as the unique solutions to the LippmaunSchwinger type integral equations 4. The action of the squareroot of the minus Laplacian on distributions Sharp estimates on the squareroot 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 squareroot of the minus Laplacian, and shall try to extend the result.
