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ON THE GENERALIZED FOURIER TRANSFORMS ASSOCIATED RELATIVISTIC SCHRODINGER OPERATORS

Research Project

Project/Area Number 09640212
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field 解析学
Research InstitutionHIMEJI INSTITUTE OF TECHNOLOGY

Principal Investigator

UMEDA Tomio  SCIENCE, HIMEJI INSTITUTE OF TECHNOLOGY PROFESSOR, 理学部, 教授 (20160319)

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)
平野 克博  姫路工業大学, 理学部, 講師 (90316034)
Project Period (FY) 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)
KeywordsSPECTRAL THEORY / SCATTERING THEORY / RELATIVISTIC SCHRODINGER OPERATORS / ELGENFUNCTION EXPANSIONS
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.

Report

(5 results)
  • 2000 Annual Research Report   Final Research Report Summary
  • 1999 Annual Research Report
  • 1998 Annual Research Report
  • 1997 Annual Research Report

Research Products

(22 results)

All Other

All Publications

  • [Publications] Tomio Umeda: "The Action of √<-Δ> on Weighted Sobolev Spaces"Letters in Mathematical Physics.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Tomio Umeda: "Eigenfunction Expansions Associated with Relativistic Schrodinger Operators"Conference Proceedings of Partial Differential Equations 2000.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 岩崎千里: "Construction of the Fundamental Solution for a Degenerate Equation and a local Version of Riemann-Roch Theorem"数理解析研究所講究録. 1156. 146-156 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Toshihiko Hoshiro: "Mourre's Method and Smoothing Properties of Dispersive Equations"Communications in Mathematical Physics. 202. 255-265 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Toshihiko Hoshiro: "On the Estimates for Helmholtz Operators"Tsukuba Journal of Mathematics. 23. 131-149 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Chris Pladdy: "Radiation Condition for Dirac Operators"Journal of Mathematics of Kyoto University. 37. 567-584 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 楳田登美男: "入門複素関数論"学術図書出版社. 160 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Tomio Umeda: "The action of √<-Δ> on weighted Sobolev spaces"Letters in Mathematical Physics. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Tomio Umeda: "Eigenfunction expansions associated with relativistic Schrodinger operators"Conference Proceedings of Partial Differential Equations 2000.. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Chisato Iwasaki: "Construction of the fundamental solution for a degenerate equation and a local version of Riemann-Roch theorem"Suuri Kaiseki Kenkyusho Koukyuroku. 1156. 146-156 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Toshihiko Hoshiro: "Mourre's method and smoothing properties of dispersive equations"Communications in Mathematical Physics. 202. 255-265 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Toshihiko Hoshiro: "On the estimates for Helmholtz operators"Tsukuba Journal of Mathematics. 23. 131-149 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Chris Pladdy: "Radiation condition for Dirac operators"Journal of Mathematics of Kyoto University. 37. 567-584 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Tomio Umeda: "Introduction to complex function theory"Gakujutsu Tosho Shuppan. 160 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 楳田登美男: "The Action of √<-Δ> on Weighted Sobolev Spaces"Letters in Mathematical Physics.

    • Related Report
      2000 Annual Research Report
  • [Publications] 楳田登美男: "Eigenfunction Expansions Associated with Relativistic Schrodinger Operators"Conference Proceedings of Partial Differential Equations 2000.

    • Related Report
      2000 Annual Research Report
  • [Publications] 岩崎千里: "Construction of the Fundamental Solution for a Degenerate Equation and a Local Version of Riemann-Roch Theorem"数理解析研究所講究録. 1156. 146-156 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 保城 俊彦: "Mourre's method and smoothing properties of dispersive equations"Communications in Mathematical Physics. 202. 255-265 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] 保城 俊彦: "On the estimates for Helmholtz operator"Tsukuba Journal of Mathematics. 23・1. 131-149 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] 楳田 登美男: "入門複素関数論"学術図書出版社. 160 (1998)

    • Related Report
      1999 Annual Research Report
  • [Publications] Chris Pladdy: "Radiation condition for Dirac operators" Journal of Mathematics of Kyoto University. 37・4. 567-584 (1997)

    • Related Report
      1998 Annual Research Report
  • [Publications] 保城 寿彦: "Mourre's method and smoothing properties of dispersive equations" Communications in Mathematical Physics. (掲載予定). (1999)

    • Related Report
      1998 Annual Research Report

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Published: 1997-03-31   Modified: 2016-04-21  

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