• Search Research Projects
  • Search Researchers
  • How to Use
  1. Back to previous page

On distribution of scattering poles for several convex bodies

Research Project

Project/Area Number 16540189
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Global analysis
Research InstitutionKyoto University

Principal Investigator

IKAWA Mitsuru  Kyoto University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (80028191)

Co-Investigator(Kenkyū-buntansha) OKAJI Takashi  Kyoto University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (20160426)
KOKUBO Hiroshi  Kyoto University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (50202057)
MORITA Takehiko  Hiroshima University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (00192782)
KAWASHITA Mishio  Hiroshima University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (80214633)
KOIKE Tatsuya  Kyoto University, Graduate School of Science, Assistant Professor, 大学院・理学研究科, 助手 (80324599)
Project Period (FY) 2004 – 2005
Project Status Completed (Fiscal Year 2005)
Budget Amount *help
¥2,000,000 (Direct Cost: ¥2,000,000)
Fiscal Year 2005: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2004: ¥1,100,000 (Direct Cost: ¥1,100,000)
Keywordswave equation / scattering matrix / modified Lax-Phillips conjecture / zeta function / system of the classical mechanics / quantum mechanics / WKB method / 散乱理論 / 修正版ラックス-ヒリップス予想 / 散乱極
Research Abstract

The main purpose of this research project is to solve the modified Lax-Phillips conjecture which is concerned with the distribution of scattering poles. To approach this problem, firstly we consider the case of three strictly convex bodies, and tried to find out a breakthrough.
In conclusion, we believe that we have found out a breakthrough. We have to go forward several steps more in order to verify our discovery. Therefore, even if it may take more than two years to arrive the final goal and have to write several preparatory papers according each step, it is sure that we have found out a new method which makes us possible to attack the modified Lax-Phillips conjecture.
A typical problem concerning the modified Lax-Phillips conjecture is the problem consider the distribution of the scattering poles for several strictly convex bodies. When we construct an asymptotic solution following WKB method, the zeta function for the classical mechanics will appear as the main part of the matrix trace of the above asymptotic solution. It is widely believed that there are close relationship between the scattering matrix and the zeta function. The most important fact of our new method is that it makes possible to treat the case of high frequencies. Indeed, when we consider quantum problems by using an approximation by the classical mechanics, the time for which the approximation is valid must be restricted within the limit of the frequency.
On the other hand, the scattering theory is concerned with the correspondance from the situation of the wave for the time near the ninus infinity to the situation of the wave for the time near the plus infinity. Thus, the limitation for the valid time makes very difficult to derive useful informations for scattering problems.
Our new idea surely opens a breakthrough for this difficulty.

Report

(3 results)
  • 2005 Annual Research Report   Final Research Report Summary
  • 2004 Annual Research Report
  • Research Products

    (15 results)

All 2006 2005 2004 Other

All Journal Article (15 results)

  • [Journal Article] Cocoon bifurcations in three dimensional reversible vector fields2006

    • Author(s)
      F.Dumortier, S.Ibanez, H.Oka
    • Journal Title

      Nonlinearity 19

      Pages: 305-328

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Annual Research Report 2005 Final Research Report Summary
  • [Journal Article] Cocoon bifurcations in three dimensional reversible vector fields2006

    • Author(s)
      F.Dumortier, S.Ibanez, H.Kokubu
    • Journal Title

      Nonlinearity 19

      Pages: 305-328

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Analyticity of the Resolvent for Elastic Waves in a Perturbed Isotropic Half Space2005

    • Author(s)
      M.Kawashita
    • Journal Title

      Math.Nachr. 278

      Pages: 1163-1179

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Annual Research Report 2005 Final Research Report Summary
  • [Journal Article] On global aspects of exact WKB analysis of operators admitting infinitely many phases2005

    • Author(s)
      T.Aoki, T.Kawai, T.Koike, Y.Takei
    • Journal Title

      Contemporary Math. 373

      Pages: 11-47

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Annual Research Report 2005 Final Research Report Summary
  • [Journal Article] Analyticity of the Resolvent for Elastic Waves in a Perturbed Isotropic Half Space2005

    • Author(s)
      M.Kawashita
    • Journal Title

      Math. 278

      Pages: 1163-1179

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On the exact WKB analysis of microdifferential operators of WKB type2004

    • Author(s)
      T.Aoki, T.Kawai, T.Koike, Y.Takei
    • Journal Title

      Ann. Inst. Fourier, Grenoble 54

      Pages: 1393-1421

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Non decay of the total energy for the wave equation with the dissipative term of spatial anisotropy2004

    • Author(s)
      M.Kawashita, H.Nakazawa, H.Soga
    • Journal Title

      Nagoya J.Math 174

      Pages: 115-126

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Existence of a singularly degenerate heteroclinic cycle in the Lorenz system and its dynamical consequences, Part I2004

    • Author(s)
      Hiroshi Kokubu, Robert Roussarie
    • Journal Title

      Journal of Dynamics and Differential Equations 16

      Pages: 513-557

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On the exact WKB analysis of operators admitting infinitely many phases2004

    • Author(s)
      T.Aoki, T.Kawai, T.Koike, Y.Takei
    • Journal Title

      Adv.in Math 181

      Pages: 165-189

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Scattering theory for the elastic wave equation in perturbed half-spaces

    • Author(s)
      M.Kawashita
    • Journal Title

      Transaction AMS (to appear)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Conley index for fast-slow systems II : Multi-dimensional slow variable

    • Author(s)
      T.Gedeon, H.Kokubu, K.Mischaikow, H.Oka
    • Journal Title

      Journal of Differential Equations (to appear)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Scattering theory for the elastic wave equation in perturbed half-spaces

    • Author(s)
      M.Kawashita
    • Journal Title

      Transaction AMS (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Conley index for fast-slow systems II : Multi-dimensional slow variable

    • Author(s)
      T.Gedeon, H.Kokubu, K.Mischaikow, H.Oka
    • Journal Title

      Journal of Differential Equations (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Scattering theory for the elastic wave equation in perturbed half-spaces

    • Author(s)
      M.Kawashita
    • Journal Title

      Transaction AMS (to appear)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Conley index for fast-slow systems II : Multi-dimensional slow variable

    • Author(s)
      T.Gedeon, H.Kokubu, K.Mischaikow, H.Oka
    • Journal Title

      Journal of Differential Equations (to appear)

    • Related Report
      2005 Annual Research Report

URL: 

Published: 2004-04-01   Modified: 2016-04-21  

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi