2000 Fiscal Year Final Research Report Summary
ON THE GENERALIZED FOURIER TRANSFORMS ASSOCIATED RELATIVISTIC SCHRODINGER OPERATORS
| Project/Area Number |
09640212
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
解析学
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| Research Institution | HIMEJI INSTITUTE OF TECHNOLOGY |
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Principal Investigator |
UMEDA Tomio SCIENCE, HIMEJI INSTITUTE OF TECHNOLOGY PROFESSOR, 理学部, 教授 (20160319)
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| Co-Investigator(Kenkyū-buntansha) |
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)
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| Project Period (FY) |
1997 – 2000
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| Keywords | SPECTRAL THEORY / SCATTERING THEORY / RELATIVISTIC SCHRODINGER OPERATORS / ELGENFUNCTION EXPANSIONS |
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| 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 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 equations
4. The action of the square-root of the minus Laplacian on distributions
Sharp 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.
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Research Products
(14 results)
