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Mathematical analysis of scattering phenomena and inverse problems

Research Project

Project/Area Number 13440048
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionTokyo Metropolitan University

Principal Investigator

ISOZAKI Hiroshi  Tokyo Metropolitan University, Graduate School of Science, Professor, 理学研究科, 教授 (90111913)

Co-Investigator(Kenkyū-buntansha) MOCHIZUKI Kiyoshi  Chuo University, Department of Science and Engineering, Professor, 理工学部, 教授 (80026773)
吉富 和志  東京都立大学, 理学研究科, 助教授 (40304729)
OKADA Masami  Tokyo Metropolitan University, Graduate School of Science, Associate Professor, 理学研究科, 教授 (00152314)
TAMURA Hideo  Okayama University, Department of Mathematics, Professor, 理学部, 教授 (30022734)
NAKAMURA Gem  Hokkaido University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (50118535)
曽我 日出夫  茨城大学, 教育学部, 教授 (40125795)
伊藤 宏  愛媛大学, 工学部, 助教授 (90243005)
倉田 和浩  東京都立大学, 理学研究科, 助教授 (10186489)
Project Period (FY) 2001 – 2003
Project Status Completed (Fiscal Year 2003)
Budget Amount *help
¥8,300,000 (Direct Cost: ¥8,300,000)
Fiscal Year 2003: ¥1,600,000 (Direct Cost: ¥1,600,000)
Fiscal Year 2002: ¥4,100,000 (Direct Cost: ¥4,100,000)
Fiscal Year 2001: ¥2,600,000 (Direct Cost: ¥2,600,000)
KeywordsScatteirng theory / Inverse problems / Dirichlet-Neumann map / Schrodinger operators / Hyperbolic space / S matrix / Electric Impedance Tomography / Numerical analysis / Electrical Impedance Tomography / 数値計算 / 弾性波 / Schrodinger作用素 / 電気伝導係数 / 散乱振幅 / ディリクレノイマン写像 / スペクトル / 複素幾何光学
Research Abstract

This reserach project aimed at the development of the study of inverse problems arising from the mathematical analysis of scattering phenoma. The main theme was the study of spectra of differential operators. As a main conference project, we have organized an international workshop of inverse problems on October 2002 at Kyoto, where leading reserchers of this field came together, and promoted much interest on this filed in Japan.
The head investigator proposed a new method for solving multi-dimensional inverse problems. The essential idea consists in embedding the inverse boundary value problem in R^n to the hyperbolic space. This new method introduced new view points of inverse problems. He discussed the inverse problem for the local perturbation of conformal metrics on hyperbolic manifolds. As an interestring by-product, he proved that in the boundary value problems in R^3, the electric conductivities can be identified locally from the knowledge of local Dirichlet-Neumann map. This hyperbolc space approach can also be applied to the problem of identification of locations of inclusions, and the reconstruction problem for linearized equations. They are expected to have applications to medical science. Okada studied numerical harmonic analysis, in particular, spline functions, wavelet analysis and numerical computation. Mochizuki studied inverse problems of reconstructing coefficients of Dirac operators from the data in finite intervals. Yoshitomi studied the properties of eigenvalues of the Laplacian on the 2-dimensional domain with crack, and also those of band region. Nakamaura studied the inverse problem of identifying the obstacle from the refelcted waves and also the inverse problem for elastic equations. Tamura studied the Aharonov-Bohm effect for the 2-diemnsional Schrodinger operators with Dirac type magnetic fields.

Report

(4 results)
  • 2003 Annual Research Report   Final Research Report Summary
  • 2002 Annual Research Report
  • 2001 Annual Research Report
  • Research Products

    (28 results)

All Other

All Publications (28 results)

  • [Publications] H.Isozaki: "Inverse spectral problems on hyperbolic manifolds and their applications to inverse boundary value problems in Euclidean space"American Journal of Mathematics. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] H.Isozaki, C.Uhlmann: "Hyperbolic geometry and local Dirichlet-Neumann map"Advances in Mathematics. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Okada: "A wavelet collocation method for property evolution equations with energy conservation"Bull Sci.Math.. 127. 569-583 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Eller, V.Isakov, G.Nakamura, D.Tataru: "Uniqueness and stability in the Cauchy problem for Maxwell and elasticity systems"Studies in Math.and its Appl.. 31. 329-349 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Mochizuki, I.Trooshin: "Inverse problem for interior spectral data of the Dirac operator on a finite interval"Publ.R.I.M.S.Kyoto Univ.. 38. 387-395 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] H.Ito, H.Tamura: "Aharanov-Bohm effect in scattering by a chain of point-like magnetic fields,"Asymptot.Anal.. 34. 199-240 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] H.Isozaki: "Inverse spectral problems on hyperbolic manifolds and their applications to boundary value problems in Euclidan space"Amer.J.of.Math.. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] H.Isozaki, G.Uhlmann: "Hyperbolic geometry and local Dirichlet-Neumann map"Adv.in Math.. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Okada: "A wave let collocation method for evolution equation with energy conservation property"Bull Sci.Math.. 127. 569-583 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Eller, V.Isakov, G.Nakamura, D.Tataru: "Uniqueness and stability in the Cauchy problem for Maxwell and elasticity systems"Studies in Math.and its Appl.. 31. 329-349 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Mochizuki, I.Trooshin: "Inverse problem for interior sectral data of the Dirac operator on a finite interval"Publ.R.I.M.S.Kyoto Univ.. 38. 387-395 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] H.Tamura, H.T.Ito: "Aharanov-Bohm effect in scattering by a chair of point-like magnetic fields"Asymptot.Anal.. 34. 199-240 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] H.Isozaki: "Inverse spectral problems on hyperbolic manifolds and their applications to inverse boundary value problems in Euclidean space"American Journal of Mathematics. (to appear).

    • Related Report
      2003 Annual Research Report
  • [Publications] H.Isozaki, G.Uhlmann: "Hyperbolic geometry and local Dirichlet-Neumann map"Advances in Mathematics. (to appear).

    • Related Report
      2003 Annual Research Report
  • [Publications] M.Okada: "A wavelet collocation method for evolution equations with energy conservation property"Bull Sci.Math.. 127. 569-583 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] M.Eller, V.Isakov, G.Nakamura, D.Tataru: "Uniqueness and stability in the Cauchy problem for Maxwell and elasticity systems"Studies in Math. and its Appl. 31. 329-349 (2002)

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Mochizuki, I.Trooshin: "Inverse problem for interior spectral data of the Dirac operator on a finite interval"Publ.R.I.M.S.Kyoto Univ.. 38. 387-395 (2002)

    • Related Report
      2003 Annual Research Report
  • [Publications] H.Ito, H.Tamura: "Aharanov--Bohm effect in scattering by a chain of point--like magnetic fields"Asymptot.Anal.. 34. 199-240 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] H.Isozaki: "Asymptotic properties of solutions to 3-particle Schrodinger equations"Communications in Mathematical Physics. 222. 371-413 (2001)

    • Related Report
      2002 Annual Research Report
  • [Publications] K.Mochizuki, I.Trooshin: "Inverse problem for interior spectral data of the Dirac operator on a finite interval"Publications of the Research Institute for Mathematical Science, Kyoto University. 38. 387-395 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] H.Soga: "Asymptotic solutions of the elastic wave equation in the care of total reflection"Communications in Partial Differential Equations. 26. 2249-2266 (2001)

    • Related Report
      2002 Annual Research Report
  • [Publications] H.Ito, H.Tamura: "Ahoronov-Bohm effect in scattering by a chain of point-like magnetic fields"Asymptotic Analysis. 21(in press). (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] J.Chang, J.Lin, G.Nakamura: "Inverse scattering for multiple obstacles"Theoritical and Applied Mechanics. 51. 401-410 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] K.Yoshitomi: "Band spectrum of the Laplacian on a slab with the Dirichlet boundary condition on a grid"Kyushu Journal of Mathematics. 57. 87-116 (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] H.Isozaki: "Asymptotic properties of solutions to 3-particle Schrodinger equations"Communications in Mathematical Physics. 222. 371-413 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] K.Mochizuki, I.Trooshin: "Inverse problem for interior spectral data of the Sturm-Liouville operator"Journal of Inverse and Ill-Posed problems. 9. 425-433 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] H.Ito, T.Tamura: "Aharonov-Bohm effect in scattering by point-like magnetic fields at large separation"Annales de l'Institute Henri Poincare. 2. 309-359 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] P.Exner, K.Yoshitomi: "Band gap of the Schrodinger operator with a strong δ-like interaction on a loof"Annales de l'Institute Henri Poincare. 2. 1139-1158 (2001)

    • Related Report
      2001 Annual Research Report

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Published: 2001-04-01   Modified: 2016-04-21  

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