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Unique continuation theorems for systems of partial differential equations and their applications

Research Project

Project/Area Number 15540164
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionKYOTO UNIVERSITY

Principal Investigator

OKAJI Takashi  Kyoto Univ., Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (20160426)

Co-Investigator(Kenkyū-buntansha) YAMADA Osanobu  Ritsumeikan Univ., College of Science and Engineering, Professor, 理工学部, 教授 (70066744)
IKAWA Mitsuru  Kyoto Univ., Graduate School of Science, Professor, 大学院・理学研究科, 教授 (80028191)
NISHIDA Takaaki  Kyoto Univ., Graduate School of Science, Professor, 大学院・理学研究科, 教授 (70026110)
SHIGEKAWA Ichiro  KYOTO Univ., Graduate School of Science, Professor, 大学院・理学研究科, 教授 (00127234)
TANIGUCHI Masahiko  Kyoto Univ., Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (50108974)
小池 達也  京都大学, 大学院・理学研究科, 助手 (80324599)
Project Period (FY) 2003 – 2004
Project Status Completed (Fiscal Year 2004)
Budget Amount *help
¥3,700,000 (Direct Cost: ¥3,700,000)
Fiscal Year 2004: ¥1,800,000 (Direct Cost: ¥1,800,000)
Fiscal Year 2003: ¥1,900,000 (Direct Cost: ¥1,900,000)
KeywordsDirac operator / Absence of eigenvalues / Limiting absorption principle / Resonance / Non-relativistic limit / スペクトル問題 / 一意接続定理
Research Abstract

The head investigator Okaji is concerned with elliptic systems of first order, especially Dirac operators that are one of the most important operators in mathematical physics. The subjects in which we are interested are spectral problems. In particular he has studied absence of eigenvalues, absolutely continuous spectrum, and resonances. He has also studied Dirac operators on non-compact surfaces.
First of all, Okaji has collaborated with H.Kalf and O.Yamada in obtaining a nice result on absence of eigenvalues of Dirac operators with diverging potentials at infinity. Moreover, Okaji has proved that the limiting absorption principle for such Dirac operators was valid. From this result, it follows that the spectrum is absolutely continuous in the whole real line. Contrary to this property it is well known that Schrodinger operators which is the limit of Dirac operators with diverging potential in the non-relativistic limit has purely discrete eigenvalues. To clarify these phenomena Okaji has investigated the behavior of resonances of Dirac type operators to show that the resonances converged to the eigenvalues of Schrodinger operators in the non-relativistic limit. He has also treated Dirac operators in a non-compact surface to show that the spectrum is purely continuous in the whole real line. This is a very meaningful result because spectrum of Laplace-Beltrami operators depends on the sign of the Gaussian curvature.
One of our investigators Yamada has collaborated with his student Ikoma in proving that Dirac aperator with Aharonov-Bohm magnetic fields has unique continuation property. He also succeeded in extending a result by Veselic (Glasnik Mat. 4, 1969) on spectral concentration of Dirac operators in non-relativistiv limit in collaboration with H.Ito.

Report

(3 results)
  • 2004 Annual Research Report   Final Research Report Summary
  • 2003 Annual Research Report
  • Research Products

    (15 results)

All 2005 2004 2003 Other

All Journal Article (11 results) Publications (4 results)

  • [Journal Article] $L^p$ multiplier theorem for the Hodge-Kodaira operator2005

    • Author(s)
      Shigekawa, Ichiro
    • Journal Title

      Seminaire de Probabilite XXXVIII, Lecture Notes in Math. 1857

      Pages: 226-246

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Annual Research Report 2004 Final Research Report Summary
  • [Journal Article] L^p multiplier theorem for the Hodge-Kodaira operator2005

    • Author(s)
      Shigekawa, Ichiro
    • Journal Title

      Seminaire de Probabilite XXXVIII, Lecture Notes Math.

      Pages: 226-246

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Orlicz norm equivalence for the Ornstein-Uhlenbeck operator2004

    • Author(s)
      Shigekawa, Ichiro
    • Journal Title

      Stochastic Analysis and Related Topics in Kyotoed (ed. by H.Kunita et al.)

      Pages: 301-317

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Chaotic composition operators on the classical holomorphic spaces2004

    • Author(s)
      Taniguchi, Masahiko
    • Journal Title

      Complex Variables 49

      Pages: 529-538

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Annual Research Report 2004 Final Research Report Summary
  • [Journal Article] The coefficient body of Bell representations of finitely connected planar domains2004

    • Author(s)
      Taniguchi, Masahiko
    • Journal Title

      J.Math.Anal.Appl 295

      Pages: 620-632

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Annual Research Report 2004 Final Research Report Summary
  • [Journal Article] Orlicz norm equivalence for the Ornstein-Uhlenbeck operator2004

    • Author(s)
      Shigekawa, Ichiro
    • Journal Title

      Stochastic Analysis and Related Topics in Kyoto (ed. by H.Kunita et al.)

      Pages: 301-317

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] The coefficient body of Bell representions of finitely connected planar domains2004

    • Author(s)
      Taniguchi, Masahiko
    • Journal Title

      J.Math.Anal.Appl. 295

      Pages: 620-632

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Orlicz norm equivalence for the Ornstein-Uhlenbeck operator2004

    • Author(s)
      Shigekawa, Ichiro
    • Journal Title

      Stochastic Analysis and Related Topics in Kyoto ed. by H.Kunita et al.

      Pages: 301-317

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Strong unique continuation property of two-dimensional Dirac equations with Aharonov-Bohm fields2003

    • Author(s)
      Ikoma, Makoto
    • Journal Title

      Proc.Japan Acad.Ser.A Math.Sci. 79

      Pages: 158-161

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Absence of eigenvalues of Dirac operators with potentials diverging at infinity2003

    • Author(s)
      Kalf, Hubert
    • Journal Title

      Math.Nachr. 259

      Pages: 19-41

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Annual Research Report 2004 Final Research Report Summary
  • [Journal Article] Strong unique continuation property of two-dimensional Dirac equations with Aharonov-Bohm fields2003

    • Author(s)
      Ikoma, Makoto
    • Journal Title

      Proc.Japan Acad.Ser.A Math.Sci. 79・9

      Pages: 158-161

    • Related Report
      2004 Annual Research Report
  • [Publications] Ikoma, Makoto: "Strong unique continuation property of two-dimensional Dirac equations with Aharonov-Bohm fields"Proc.Japan Acad.Ser.A Math.Sci.. 79・9. 158-161 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Kalf, Hubert: "Absence of eigenvalues of Dirac operators with potentials"Math.Nachr.. 259. 19-41 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Taniguchi, Masahiko: "Size of the Julia set of structurally finite transcendental entire function"Math.Proc.Cambridge Philos.Soc.. 135・1. 181-192 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Aoki, Takashi: "On the exact WKB analysis of operators admitting infinitely many phases"Adv.Math.. 181・1. 165-189 (2004)

    • Related Report
      2003 Annual Research Report

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Published: 2003-04-01   Modified: 2016-04-21  

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